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WORKS ON 

DESCRIPTIVE GEOMETRY, 



AND ITS APPLICATIONS TO 



ENGINEERING, MECHANICAL AND OTHER INDUSTRIAL DRAWING. 

By S. EDWARD WARREN, C.E. 



I. ELEMENTARY WORKS. 

1. Primaey Geometry. An introduction to geometry ac 
Qsually presented ; and designed, first, to facilitate an earlier 
beginning of the subject, and, second, to lead to its graphical 
applications in manual and other elementary schools. With 
numerous practical examples and cuts. Large 12mo, cloth, 80c. 

2. Free HAND Geometrical Drawing, widely and variously 
useful in training the eye and hand in accurate sketching of plane 
and solid figures, lettering, etc. 12 folding plates, many cuts. 
Large 12mo, cloth, $1.00. 

3. Drafting Instruments and Operations. A full descrip- 
tion of drawing instruments and materials, with applications to 
useful examples ; tile work, wall and arch faces, ovals, etc. 7 
folding plates, many cuts. Large 12mo, cloth, $1.25. 

4. Elementary Projection Drawing. Fully explaining, in 
six divisions, the principles and practice of elementary plan and 
elevation drawing of simple solids ; constructive details ; shadows ; 
isometrical drawing ; elements of machines ; simple structures. 
24 folding plates, numerous cuts. Large 12mo, cloth, $1.50. 

This and No. 3 are especially adapted to scientific, preparatory, 
and manual-training industrial schools and classes, and to all 
mechanics for self-instruction. 

5. Elementary Perspective. With numerous practical 
examples, and every step fully explained. Revised Edition 
(1891). Numerous cuts. Large 12mo, cloth, $1.00. 



6. Plane Pkoblems on the Point, Straight Line, and Circle. 
225 problems. Many on Tangencies, an d other useful or curious% 
ones. 150 woodcuts, and plates. Large 12mo, cloth, $1.25. 

IL HIGHER WORKS. 

1. The Elements of Descriptive Geometry, Shadows 
AND Perspective, with brief treatment of Trehedrals ; Trans- 
versals; and Spherical, Axonometric, and Oblique Projections; and 
many examples for practice. 24 folding plates. 8vo, cloth, $3.50. 

2. Problems, Theorems, and Examples in Descriptive 
Geometry. Entirely distinct from the last, with 115 problems, 
embracing many useful constructions ; 52 theorems, including 
examples of the demonstration of geometrical properties by the 
method of projections ; and many examples for practice. 24 fold- 
ing plates. 8vo, cloth, $2.50. 

3. General Problems in Shades and Shadows, with 
practical examples, and including every variety of surface. 15 
folding plates. 8vo, cloth, $3.00. 

4. General Problems in the Linear Perspective of 
Form, Shadow,' and Eeflection. A complete treatise on the 
principles and practice of perspective by various older and recent 
methods ; in 98 problems, 24 theorems, and with 17 large plates. 
Detailed contents, and numbered and titled topics in the larger 
problems, facilitate study and class use. Eevised edition, correc- 
tions, changes and additions. 17 folding plates. 8vo, cloth, $3.50. 

5. Elements of Machine Construction and Drawing. 
73 practical examples drawn to scale and of great variety ; besides 
30 problems and 31 theorems relating to gearing, belting, valve- 
motions, screw-propellers, etc. 2 vols., 8vo, cloth, one of text, 
one of 34 folding plates. $7.50. 

6. Problems in Stone Cutting. 20 problems, with exam- 
ples for practice under them, arranged according to dominant 
surface (plane, developable, warped or double-curved) m each, and 
embracing every variety of structure ; gateways, stairs, arches, 
domes, winding passages, etc. Elegantly printed at the Riverside 
Press. 10 folding plates. 8vo, cloth, $2.50. 



ijstdusteial science deawikg. 



ELEMENTARY 



LINEAR PERSPECTIYE 



OF 



FORM AND SHADOW. 



COMPLETE IN ITSELF, WITH THE NECESSARY PRELIMINARY PROBLEMS 
IN PROJECTIONS, AND EXAMPLES FOR PRACTICE. 

FOR DRAFTSMEN, ARTISTS, AND INDUSTRIAL DRAWING COURSES 
IN PRErARATOR><Al^'D HIGH SCHOOLS, ETC. 

J 

3n ®tDo J3arts. 

PART I.— PRIMITIVE METHODS; WITH AN INTRODUCTION. 

PART II.— DERIVATIVE METHODS; WITH SOME NOTES ON AERIAL PERSPECTIVE. 



/ 



By S. EDWARD WARREN, O.E., 

\\ 

FORMER PROFESSOR OP DESCRIPTIVE GE03IETRY, ETC., IN TUB RENSSELAER POLTTECHNIC 
INSTITUTE, ETC., ETC., AND AUTHOR OF A SERIES OF ELEMENTARY AND HIGHER 
TEXT-BOOKS ON MECHANICAL INDUSTRIAL DRAWING, AND DESCRIP- 
TIVE GEOMETRY AND ITS APPLICATIONS. 



REVISED AND ENLARGED EDITION. 




NEW YORK: \. V^^^ohTngT< 

JOHN WILEY & SONS, ' --? c^- 

53 East Tenth Street. 
1891. 






Copyright, 1891. 
By S. EDWARD WARREN, 




CONTENTS. 



PAGE 

Preface vi 

INTRODUCTION. 

Chapter I. — Instruments, Materials, and Methods 9 

Paper. Support of Paper. Pea oils. Rulers 9 

Compasses. Irregular Curves. Indian Ink 10 

Construction and Execution of Drawings , 11 

Chapter II. — Preliminary Principles and Explanations 12 



PART I. 

primitive methods. 

Chapter I. — Defiyiitions and General Principles 1 

" II. — Tlie Elements of P'ojection^ 1 

" III. — The Construction of the Perspectives of Objects from, their Pro- 
jections. Examples for Practice 24 

" IV. — Pml Projections, and Perspectives made from them 28 

Perspectives of Geometrical Solids, Art. (68.) 32 

Example 1. — To Find the Perspective of a Vertical Square 

Prism 32 

" 2. — To Find the Perspective of a Triangular Pyra- 
mid. Examples for Practice 35 

Chapter Y,— -Removal of Practical Difficulties, arising from the Confusion of 

Projections and Perspectives 36 

§ I. — First Method. — Translation forward of the Perspective Plane. . 36 

Example 3. — To Find the Perspective of a Cube, etc 38 

§ 11.— Second 3Iei7iod.—Vse of Three Planes 38 

Example 4.— To Find the Perspective of an Obelisk, etc 41 

Chapter VI. — Projections and Perspectives of Circles, arid of Bodies having partly 

or wholly Curved Boundaries 43 



11 CONTENTS. 

PAGE 

Example 5. — To Find the Perspective of a Circle, lying in 

the horizontal plane 43 

Of Planes, Arts. (77-84.) 45-46 

" 6. — To Find the Perspective of a Cylinder, stand- 
ing on the horizontal plane 47 

** 7. — To Find the Perspective of a Cone, standing 

on the horizontal plane 50 

" 8. — Do. of a Cone whose axis is parallel to the 

ground line . 52 

** 9. — Do. of a Cone whose axis is parallel to the 

vertical plane only 53 

** 10. — Do. of a Cone whose axis is oblique to both 

planes of projection 54 

" 11.— To Find the Perspective of a Sphere 58 

First Method of finding the apparent contour 58 
Second Method " " " 61 

** 12. — To Find the Perspective of a Concave Cupola 

Koof 63 

Examples for Practice 64 

Chapter VII. — Perspectives of Shadows 65 

General Principles and Illustrations, Arts. (89-99.) 65 

Example 13. — To Find the Perspective of the Shadow of a 

Square Abacus on a Square Pillar 67 

** 14. — Do. of a Triangular Pyramid upon the Hori- 
zontal Plane 69 

** 15. — Do. of a Dormer Window upon a Roof 71 

16. — Do. of a Convex, Four-sided Dome, Flag- 
staff, and their Shadows 73 

Examples for Practice 74 



(( 



PART II. 

DERIVATIVE METHODS. 

Chapter I. — General Principles and Illustrations 75 

Example 1. — To Find the Vanishing Point of Telegraph 

Wires, etc 78 

" 2. — To Find the Vanishing Point of a Perpendic- 
ular and of a Diagonal 79 

" 2a.— To Find the Perspective of a Perpendicular 

and a Diagonal 80 

Examples for Practice 81 

Particular Derivative Methods, Arts. (118-121.) 81 



CONTENTS. iii 

FAGB 

Example 3. — To Find the Perspective of a Straight Line, in 
any position oblique to both planes of pro- 
jection, etc 82 

Examples for Practice 83 

*' 4. — To Find the Perspective of a Tower and Spire. 84 

Practical Remarks, Art. (123.) 86 

" 5. — To Find the Perspective of a Cross and Pedestal 89 

Examples for Practice 91 

Chapter II. — Ih'spectivts of Shadows 92 

Example 6. — To Find the Vanishing Point of Rays, and of 

their Horizontal Projections 92 

" 7. — To Find the Perspective of the Shadow of any 

Vertical Line upon the Horizontal Plane . . 93 

" 7a.— To Find Shadows on Side-walls 95 

Chapter III. — ^fiscellaiuoics Problems 96 

Example 8, — To Find the Perspective of a Pavement of 
Squares, whose sides are parallel to the 

ground line 96 

" 9. — Do. of a Pavement of Hexagons, whose sides 
make angles of 30° and 90° with the ground 

line 98 

" 10.— Do. of an Interior 98 

** 11. — Do. of the Shadows in an Interior 102 

" 12.— Do. of a Cabin 104 

" 13. — Do. of the Shadow of a Chimney on a Roof.. 108 

Examples for Practice 109 

Chapter IV. — Pictures, and Aerial PRrspeetive 110 

Landscape Outlines 110 

Landscape Details.— Trees. Hills. Valleys Ill 

Ascent and Descent. Level of the Eye 112 

Reflections in Water. Location of the Centre of the Picture 113 
Location of the Perspective Plane. Shadows of Trees, and 

other Vertical Objects. Time of a Given Aspect 114 

Light and Shade 115 

Edges. Color 116 



APPENDIX. 

§ 1. — Description of Plates 117 

§ 2. — Generalization of the Horizontal Plane and Horizon 119 

§ 3, — Perspectives from Perspective Plans 121 



To instruct the young in the principles of geometry is to render them the 
greatest service. Not only does this science develop the intelligence, but it 
makes precision habitual, thanks to the exact knowledge which it gives of the 
dimensions of all sorts of bodies, considered under different aspects. With- 
out knowledge of geometry, the artist, uncertain of his result, works blindly. 
Led by ignorance, he drags the young confided to his care from error to 
error. — From Baclielier's ' ' Memoir on the Boyal Free Drawing School of 
Paris" written a hundred yea/rs ago. 



1 



PKEFACE. 



For several years, while teaching a comparatively advanced 
course on Perspective, I purposed to compose an elementary 
perspective, which should be interesting and popular, without 
being empirical ; and, on the other hand, perfectly demonstra- 
tive, without being too elevated. In other words, I have sought 
to make my work elementary, not in the sense of merely stating 
perspective facts without adequate explanation, but in the sense 
of selecting simple, yet widely and always useful examples, and 
then fully explaining, in easy order, the few plain principles 
necessary to the solution of such examples. 

An exact knowledge of perspective is indispensable to those 
who would make exact representations, for industrial purposes, 
of architectural or mechanical structures as they appear. It is 
highly useful to those, even, who practise perspective as an 
ornamental art, in the making of pictures ; inasmuch as it 
enables them to know scientifically, as well as feel sensibly, 
whether their drawings are correct or incorrect. It is also 
interesting to the amateur judges and admirers of pictures, as 
well as to their makers ; and, finally, it is useless to none who are 
in any manner engaged with the arts of graphical representation 
or design. 

It is a part of the price of truth, w^hereby we discover its 
worth, that we must discover the truth concerning propriety of 
arrangement in any subject, as an indispensable condition for 
its successful treatment. 

'' Projections/' which, usually for industrial purposes, repre- 



VI PREFACE. 

sent objects as they are, in form and size, naturally precede, in 
a course of exact study, ^' Perspective," which, usually for pic- 
torial effect, represents objects as they appear. Perspectives, or 
drawings of objects as they appear, are made, then, from Projec- 
tions, or drawings of objects as they are; and which, therefore, 
are competent representatives of those objects. 

The study of projections thus properly preceding that of 
perspective, as its natural foundation, disadvantages will una- 
voidably arise from attempts to treat of exact perspective, 
without a formal preliminary treatment of projections. Hence, 
this work, while complete in itself, is the natural successor, for 
those who use both, of my ^^ Elementary Projection Drawing/' 
in which objects are shown in projection only. 

It is unfortunate for learners that a subject so simple, useful, 
and attractive, as Perspective is, when properly treated, should 
come to be regarded with aversion, merely owing to defects in 
its treatment ; the chief of which defects is, perhaps, the failure 
fully to exhibit its foundation in ^^ projections.''^ 

The present volume is an attempt to expressly present Per- 
spective, as founded on ^^ Projections," and in this respect it 
differs, more or less noticeably, from numerous elementary 
works on the subject. - I accordingly hope for such results, in 
respect to ready and interested understanding of the subject, 
as the improved treatment of it, which I have endeavored to 
give, leads me to anticipate. 

Only a few, and quite simple^ problems in perspectives of 
shadows have been inserted. Yet they are believed to be suf- 
ficient. 

The tried and proved feature, so obviously important in text- 
books of every grade, and in every branch, of applied mathemat- 
ics, of examples for practice, additional to those explained in 
the text, has been incorporated in this as in my other volumes, 
generally. 

The simplest conception, and resulting definition, of the per- 



PREFACE. Vll 

spective of a point is, that it is where the " visual ray " through 
the point pierces the plane of the picture ; i.e., the "perspective 
plane." 

The method of construction just indicated, and adopted for 
Part I., does away with the whole machinery of "vanishing 
points/' "perpendiculars," "diagonals," etc.; and, accord- 
ingly, these, with their advantages, are briefly explained and 
illustrated in Part II., giving rise to derivative methods of 
construction, and aiding the reader in understanding the meth- 
ods usually employed by writers on perspective. 

In my larger work on " Higher Perspective," and to some 
extent, also, in my " Elements of Descriptive Geometry, Shad- 
ows, and Perspective," I have endeavored to exhibit more fully 
a systematic arrangement of all its methods, and interesting 
peculiarities and details. 

In a brief Appendix will be found two plates, indicating such 
work as a mastery of the principles of this volume will enable 
one to execute ; also, a few hints which may aid the student in 
his further practice. 

Finally, while the subject, as here treated, has been, and may 
still be, sometimes found sufficient for the purposes of Engineer- 
ing, or other Technical Schools, yet, with the greatly increased 
attention given to preparation for skilled industrial life, this 
volume is intended to be suited to the wants of Preparatory 
Scientific Courses in Academies, High Schools, etc. 

Newton, Mass., 1891. 



NOTE TO TEACHERS. 



While no one method of using this volume can be declared to be abso- 
lutely the best for all cases, yet it is suggested to teachers, as the result of 
long experience in the use of this and the other volumes on kindred subjects, 
that a very efficient method of doing the most and the best work, in a given 
time, is to have exercises of mingled interrogation, or blackboard demonstra- 
tion and drawing, from one to two hours in length, according to the grade 
of the class ; which may most conveniently consist of twelve to twenty work- 
ing together at once — each at his drawing, except while called to explain a 
principle, or point of construction, from his drawing, or at the blackboard. 



LINEAR PERSPECTIVE. 



INTRODUCTION, 

CHAPTER I. 

INSTRUMENTS, MATERIALS, AND METHODS. ' 

1. Paper. — For elementary practice, thick unruled writing, or 
tough printing paper, will answer. For nicer work,- German 
cartoon, or English smooth drawing paper, will be convenient ; 
and for exact constructions, in lines or tints of Indian Ink, 
Whatman^s drawing paper will be best. 

2. Support of the Paper. — For slight pencil or ink sketches, 
the paper may lie flat on an atlas, or any firm, but not rigid sur- 
face. For larger and exa^t ink drawings, the paper must be 
well wet, and then fastened round the edges with gum-arabic, 
or mucilage, to a smooth and accurately square-cornered drawing 
board. When dry, it will be found to be tightly stretched. 
Care should be taken to keep paper flat and smooth, when not 
stretched as just described. 

3. Pencils. — For sketching, use a hard pencil, as No. 4, or 5, 
of Faber's, and make only the faintest lines. For finishing up 
pencil drawings, use softer and blacker pencils, as InTo. 3 for 
well defined objects, and Nos. 1 and 2 for shadows, foliage, &c. 

In pencilling a drawing which is to be inked, use a pencil 
sharpened on a fine file, to a thin edge, rather than a round 
point, since it will thus keep sharp much longer. 

4. Rulers. — For drawing on stretched paper, use a T rule, for 
drawing all lines from right to left ; and a right angled triangle, 
for drawing lines perpendicular to these. With loose paper^ use 
a common ruler and right angled triangle. 



10 LINEAR PERSPECTIVE. 

5. To draw parallels in any oblique position^ by a ruler and tri 
angle. To draw through p^ for example, a parallel to ah. Place 




a side of the triangle on ah^ and bring up the ruler ac, as shown 
in Fig. 1. Hold the ruler fast, and slide down the triangle to the 
position (2) when pd will be parallel to ah. 

6. To draw perpendicidars in ohlique positions. — Slide the tri- 
angle, as before, till de passes through jo, then d being a right 
angle, a line can be drawn through p^ and perpendicular to ah. 

7. Compasses. — For drawing ink or pencil circles, the com- 
passes should have movable legs, which may be replaced by a 
drawing pen, or pencil-holder. 

8. In using the compasses^ hold them by the joint, with the 
thumb and forefinger. Then, in setting off distances on a line, 
turn them, alternately, on one side and the other of the line, never 
taking both points at once from the paper, till the operation is 
finished. This method is most expeditions and accurate. Like 
wise, in describing a circle, the whole can be accomplished with 
quick and uninterrupted motion. 

9. Irregular Curves. — For drawing other curves than circles, 
points of which have been previously constructed, use the thin 
plate of wood with variously curved edges and openings, and 
called an irregular curve. 

10. Indian Ini^. — This color, when of good quality, is of a 
brownish black, and is prepared in polished or gilded cakes, fine- 
grained, and usually scented with musk or camphor. It is prepared 
for use, like other water colors, by touching the end to water, an^^ 



INSTRUMENTS AND MATERIALS. 11 

rubbing on an earthen plate or tile. When enough has been 
ground off, wipe the cake dry to prevent its crumbling. 

This ink, when thick, may be applied in a drawing-pen, or 
brush, so as to make black lines, or surfaces. When diluted 
with a quantity of water, tints of any degree of lightness may be 
quickly laid on the paper (stretched) by a rapid use of a goose- 
quill sized, or larger camel's hair brush. 

11. Drawings made by students, both for gaining and illus- 
trating a knowledge of principleSf should retain in ink all the 
auxiliary projections from which the perspectives are made. 
But in actual practice, and in the plates made by students for 
pictorial effect, where the perspective alone is intended to occupy 
the given sheet or plate, these auxiliary figures should be pen- 
cilled only, on separate sheets ; which may conveniently be of 
buff Manila office paper, temporarily attached to the picture 
sheet, while the perspective figure on the latter is drawn. 

By methods quite artificial, and useful in advanced practice, 
but not obvious to a learner, and therefore increasing the liabil- 
ity to error and hinderance,* these auxiliary constructions can 
be wholly or partly dispensed with. But the elementary methods 
of this volume, in using which error is impossible with ordinary 
care, are recommended, by preference, for common use as suffi- 
cient and satisfactory. 

\\a. The finished perspectives mentioned in the last article 
are finished in various ways. Objects of an architectural char- 
acter being those most frequently represented, they may, espe- 
cially when of masonry, be finished in free-hand pen work 
(see Plate I., Fig. 1), or in India-ink, sepia, or in colors, with 
more or less of landscape accessories. It is thus a great advan- 
tage to the perspective draughtsman to have some practical ac- 
quaintance with free-hand sketching, and water-color tinting 
and drawing, f 

* Examples of which may be found in my " Higher Perspective "; and in my 
"Elements of Descriptive Geometry, Shadows, and Perspective." 

t See my " Elementary Geometrical Free-Hand Drawing," and '' Drafting In- 
Btruments and Operations"; also, any practical works on Topographical and 
Water-color Drawing. 



12 LIJfEAE PERSPECTIVE. 



CHAPTER n. 

PEWJMINARY PRINCIPLES AND EXPLA N^ATIONS. 

12. Sitting by a window, you may fix your attention on all 
that you see through one of its panes — buildings and parts thereof, 
trees, roads, fields, woods, streams and clouds. 

As soon, however, as you give attention, both to the pane and 
to what you see through it, you will find that, by looking with each 
eye separately, you will see partly different sights through the 
same pane. Hence, to see definitely both the pane and what you 
see through it, you must close one eye. 

13. This being done, you might paint upon the glass everything 
that you see through it, just where you see it, and of the same 
shade and color. A perfect pictui'e, in every respect, of all seen 
through the glass, from one point of sight, might thus be made on 
the pane. Such a picture would be called the perspective of the 
view seen through the pane. 

14. I. Hence ^ perspective \?> a picture which shows one or more 
objects just as they appear, in respect both to form and color, and 
as seen from one fixed point of sight. See Fig. A (24a). 

15. If seated quite near the window, you will observe that you 
cannot see all that is to be seen through it, without turning the 
head ; while each new direction of sight gives you at least a partly 
new view. Also each new position of the eye gives, evidently, a 
different view through the same pane. 

16. H. Hence any single perspective drawing should embrace 
no more than one view, that is, no more than can really be seen 
when looking in one direction from one fixed point of sight. 

17. The chief exceptions to this rule are in panoramic and 
architectural interior scene painting, w^hich, being intended to 
please large assemblies, are painted from several points of sight, or 
from one quite remote one. 

18. All this being understood, suppose you are in a field, and 
dewing a distant tree through a framed pane of glass, held at a 

axed distance from the eye. As you approach the tree it appears 
to occupy a larger and larger portion of the glass ; while, as you 
recede from it, a contrary effect is produced. 



PRELIMINARY PRINCIPLES AND EXPLANATIONS. 13 

in. Hence the size of the object, in a picture^ depends on its 
distance from the eye. See Figs. A and B (24«). 

19. Again; if your distance from the tree is fixed, the nearei 
the pane is carried to the tree the more completely will the view 
of the tree fill it. That is— 

IV. The comparative size of an object in the picture, and the 
whole picture, depends also on the distance of the picture plane 
from the object. See Fig. A (24a). 

20. Further, if two trees, at equal distances, and of difierent 
sizes, be viewed at once through the same pane, and from the same 
fixed position, the larger one will cover a larger space on the pane, 
as seen through it. 

V. Hence, other things being the same, the size of an object, 
in the picture, depends on its actual size. 

21. Once more, by moving, together with the pane, from side 
to side, or up and down, the tree will be seen through diiferent 
portions of the pane, when seen from the different positions so 
taken. 

YI. That is, the place of an object in a given picture, its size 
and distance being also given, depends on its direction fiom the 
observer. 

22. VII. From the last four particulars, w^e now conclude that, 
in order to represent a given object truly, its dimensions^ distance 
from the picture^ distance from, the eye, and direction must all be 
known. 

In other words, the relative position of the eye, the picture^ and 
the object^ and the size of the latter^ must be known. 

23. Returning now to the picture painted on the window pane, 
each point of that picture is in a straight line, from the point repre- 
sented, to the eye. Such a line is called a visual ray. 

VIII. Hence the perspective of any point is where the visual 
ray from that point meets the surface of the picture. 

Finally, the following general principles may serve to connect 
his introductory sketch, which embraces the primary facts of per 
spective, given by the testimony of the senses, with the more exact 
treatment of the subject, which succeeds, and in which the prin- 
ciples of perspective, based upon these facts, are demonstrated. 

24. Science is a complete body of truth, whose parts are naturally 
related to each other ; and hence may be expressed by a systematic 
and connected statement of successive particulars, proceeding in 
natural order from primary elements to complete results. 

Perspective science is such a body of truth, relating to the 



14 



LINEAR PERSPECTIVE. 



manner of representing objects as they appear. This science is 
founded on the simple facts of vision already described, and 
which are learned by observing what and Jioto we see. 

24^?. Let MT, Fig. A, represent a maple-tree, E the eye, or 
point of sight, and PQ the perspective or picture plane. Then, 
ME and TE being the visual rays (23) from the foot and top of 
the tree, mt is the perspective (14) of the tree on the plane PQ. 

Again, PQ, Fig. B, is at the same distance from the eye as 
in Fig. A, but is nearer the tree MT. Hence (18) the perspec- 
tive, mt, both is and appears larger than mt in Fig. A. It is 
larger, because PQ is nearer the tree. It appears larger, because 
the eye is nearer the tree, and therefore the visual angle, MET, 
which determines the apparent size of the object, is larger than 
in Fig. A. 

Finally, in Fig. A, while the picture m't' is larger than mt 
(19), it does not appear larger to the eye at E, because, the dis- 
tance of the eye from the tree being unchanged, the size of the 
visual angle MET is unchanged. 

We now proceed to unfold the elements of Perspective from 
the preceding simple facts of vision, and to apply those ele- 
ments to practical exercises in perspective drawing. 




^-E 



— M 



Fig. B 



DEFINITIONS AND GENERAL PKINCIPLES. If 



PAET I. 

PRIMITIVE METHODS, 



CHAPTER I. 

DEFINITIONS AND GENERAL PRINCIPLES. 

26. The complete perspective of an object, is a picture of it, 
which, when viewed from a certain point, produces the same image 
upon tlie eye that the object itself does, when viewed from the 
same point. 

Each point and line of such a picture, must, when suitably placed 
between the eye and the object, exactly cover and conceal from 
view the corresponding points and lines of that object. It must 
also, as truly as art will allow, present,, at each point, the same 
shade and color that is exhibited by the same object. 

26. Hence perspective embraces two branches: the persjoective 
of /brm, called linear perspective ; and the perspective oi color and 
gradations of shade^ called atrial perspective. 

27. Aerial perspective is an imitative art^ founded- on extensive 
observation of nature, and on the science of optics. 

Linear perspective is either an imitative art, or an art of exact 
geometrical construction, according as the outlines of pictures of 
given objects are traced by the eye, or constructed with instru- 
ments, according to geometrical principles. 

28. In point of fact, linear perspective is practised as a construc- 
tive art, chiefly in its application to regular objects. It is practised 
as an imitative art, mainly in the drawing of irregular, or pic 
turesque objects, such as trees, animals, hills, streams, and old 
buildings. 

We will next inquire into the natural principles, which lead to 
the exact construction of the perspectives of objects. 

29. The eyes are so related, that in attempting to see, with both 
of them together, objects at different instances, distinctly and at 
once, w^e see these objects partly double (12). 

Hence in making an exact picture of any object, we suppose it 



IC LINEAR PERSPECTIVE. 

to be Viewed with one eye, or that the two eyes are reduced to a 
single seeing point, called the point of sight (14). 

30. Objects become visible by means of rays of light, reflected 
from them to the eye, and called visual rays (23). 

31. Rays of light proceed in straight lines ; as is proved by the 
fact that we can see nothing through an opaque bent tube. 

32. The visible boundary of an object is called its apparent con- 
tour. The perspective of this contour, is the linear perspective of 
the object. 

33. Any body, having a vertex, and plane sides, is, in a generii' 
sense, called a Pyramid. 

Any curved surface, having a vertex, and therefore containing 
straight lines drawn through that vertex, is a Cone^ in the general 
sense of the term. 

Hence visual rays, from all points of the apparent contour of an 
object to the eye, form a pyramid.^ or a cone — whose vertex is the 
point of sight — according as the object is bounded by straight or 
by curved lines. This being understood, this pyi-amid and cone 
are, for the sake of brevity, called indifferently the visual cone. 

34. Next, conceive a plane to intersect the visual cone, anywhere 
between its vertex, that is the eye, and its base, that is the object. 
This plane will cut from the cone a figure which will exactly conceal 
from the eye at its vertex, the apparent contour of the original 
object. That is (25) this figure will be the linear perspective (27) 
of that contour, and hence of the object (32). 

35. In like manner, the intersection of the visual ray (23) from 
any one point of the given object, with the given plane, is the 
perspective of the point from which that ray proceeded. 

36. The given plane is therefore called the perspective plane ^ 
and is understood to be vertical, unless the contrary is mentioned. 

Illustration. — In Fig. 2, let E be the position of the eye, ABC 
the object, as a wood or paper triangle, to be represented ; and PQ, 
the perspective plane. Then AE, BE, and CE, represent visual 
rays from the corners or vertices of the given triangle. Now let 
a, 5, and c represent the points in which these visual rays jnerce 
the perspective plane PQ, then ahc will be the perspective of ABC. 

37. It now clearly appears, that, in order to find the perspective 
of an object, three things must be given ; the object itself, the posi- 
tion of the eye^ and the perspective plane (22). Observe here, also, 
that, as lines from the visible points of the object to the eye are 
visual rays, these rays become known as soon as the positions of the 

I eye and of the object are given. 



DEFINITIONS AND GENERAL PRINCIPLES. 



!• 



If either of the lines, as a5, of the perspective, M'cre prolonged 
either way, or both ways, it would be called the indefinite 'perspec- 
tive of the original line as AB. 

38. Any angle, as AEC, formed at the eye by two visual rayt* 



A«-- 





.^^ 

s 



Fig. 2. 

is called a visual angle. It now appears that any line, as AC, and 
its perspective, ac, subtend the same visual angle. The reason, 
therefore, why an object and its perspective present the same 
appearance (25) to the eye, is, that they subtend the same visual 
angle ; for the apparent size of any object depends on the size of the 
visual angle which includes it. Hence if two equal lines be in 
parallel positions, but at unequal distances from the eye, the further 
one will subtend the smaller visual angle and will therefore appear 
the shorter (18). Also, if a line or a surface be viewed obliquely, 
instead of directly, it will appear of diminished size, and is said to 
he foreshortened. 

39. The position, E, of the eye, and the form and position of the 
object ABC remaining fixed, there will be as many different sizes 
and forms of the perspective, abc, as there may be different dis- 
tances and positions of the perspective plane, between E and ABC 
And these various forms and sizes of abc will all be true perspec- 
tives of ABC. To understand this completely, it is only necessary 
to remember, 1°: That all these forms of abc are sections of the 
same visual pyramid ABC— E (34), and 2*=* : That the definition of a 
perspective is not, a figure that is as the original object appears; 
but, only, one that appears as that objett does, when viewed from 
the sane point (25). 

2 



IB 



LINEAR PERSPECTIVE. 



CHAPTER n. 



THE ELEMENTS OF PROJECTIONS. 



40. It is evident from a consideration of Fig. 2, that we cannot; 
practically, find a perspective picture directly according to (35) ^. e. 
directly from objects themselves. Visual rays are invisible and 
intangible, and we cannot conveniently substitute for them, threads 
from every point of an object, as a house, to a fixed point, as the 
top of a stake, taken to represent the place of the eye, and then 
find where all the threads pierce a paper plane, set up between the 
object and the place of the eye. 

41. What then can be used in place of the actual object, from 
which to make its perspective, as truly as if found mechanically, as 
above described? We employ auxiliary drawings, which show 
the positions, forms, and dimensions of the original objects, just a^? 
they really are, and from such drawings, with similar representa- 
tions of the visual rays, we construct the perspectives, which shovv 
those objects as they appear, 

42. These auxifiary drawings, which show the given object, and 

its visual rays, as they really 
are, in respect to form and 
relative position, are called 
projections. To the expla- 
nation and construction of 
projections, we therefore 
turn, as the next thing in 
order. 

43. lUicstration. — Let 
HH', Fig. 3, represent a 
level plane, called the hori- 
zontal plane, and VV an 
upright plane, at right an- 
gles to IIH', and hence; 
called the vertical plane. 




Fig. 3. 



The floor and any wall of a room, would be such a horizontal and 



THE ELEMENTS OF PROJECTIONS. 19 

vertical plane. HV, the intersection of these planes, is called the 
ground line. 

Next, let P be any point in the open angular space between these 
two planes. Then let Pjt? be a straight line from P, perpendicular 
to the horizontal plane, HH', and meeting it at some point repre- 
sented by p. Likewise, let Yp' be a line from P, perpendicular to 
the vertical plane, YV, and meeting it at p' . Then P is called 
the horizontal projection of P, andjo'is the vertical projection of P. 

44. Observe now, according to (41) that the two projections of a 
point are an adequate representative of the real position of the 
point. For ^'5', the vertical height of the vertical projection, jt>', 
above the ground line HV, is equal to the real height, Pjo, of the 
point P above the horizontal plane. Likewise pq^ the perpendicu- 
lar distance of the horizontal projection, jt), from the ground line, is 
equal to the real distance, Pjo', of the point in space, P, from the 
vertical plane. Hence a point is named by naming its projections / 
thus, we describe P as the point j^p'. 

45. If a point, as S, is in the horizontal plane, it coincides with 
its horizontal projection, 5, and its vertical projection, s\ must be in 
the ground line HV. Likewise, if a point, as R, lies in the vertical 
plane, it is its own vertical projection, r', and its horizontal projec- 
tion, r, must be in the ground line. 

46. By considering the explanations just given, we are led to the 
following additional practical particulars. First : The forms of 
bodies are indicated by the positions of the points which compose 
or limit their bounding lines. Hence, if the distinguishing points 
if the boundaries of an object be projected in the simple manner 
just explained, and if the projections of these points be connected, 
m each plane, the projections of the object will be formed. 
Second: p represents the real point, P, as it would appear if seen 
from above, in the vertical direction Pjt?. Likewise p' represents 
the same point as seen in looking in the direction Vp'. Third: Li 
order to view all the points of an object simultaneously in the same 
direction, the eye must be at an indetinitely great distance from it. 
Hence projections represent objects as they would appear, if visible 
from an indefinitely great distance, and viewed in a direction per- 
pendicular to each plane of projection in succession. 

47. Illustration. Fig. 4. GL is the ground line ; GH, the hori- 
zontal plane ; and GV, the vertical plane. ABC-D is a triangular 
prism, placed with its edges parallel to the ground line. In view- 
ing this prism from a great distance above it, so as to look at all 
parts at once, in the parallel directions Aa, F/*, &c., the two sloping 



20 



LINEAR PERSPECTIVE. 



sides, ACDF and ABDE, will be visible. Projecting the corners 
of these faces by vertical projecting lines, Aa, &c., as in Fig. 3, we 
find acfd for the horizontal projection of ACFD, and dbde for the 
horizontal projection of ABDE. 




Fig. 4. 

Likewise, in viewing the prism in the direction Aa', its front, 
ACDF, only, is visible, and a' o' d'f is the vertical projection of 
this face. 

It thus appears that the horizontal projection shows the true 
relative distances of all points of the prism, front of the vertical 
plane ; and that the vertical projection shows the true heights of all 
points of the prism above the horizontal plane. That is, the two 
projections, together, form an adequate representative of the form 
of the prism. 

48. In particular, the face BCEF is parallel to the horizontal 
plane, and hcef^ its horizontal projection, is equal to it. That is, 
when a surface^ or line, is parallel to a plane of projection, its true 
size is shown in its projection upon that plane. Again, CB is a 
line which is perpendicular to the vertical plane, and hence the 
point c' is its vertical projection. That is, when a line is perpen- 
dicular to either plane of projection, its projection on that plane ia 
a point. 

49. Continuing to examine this figure, 4, with reference, now, to 
its lines only, it appears that CF, for example, is parallel to both 
planes of projection, and hence to the ground line, and its projec- 
tions c/'and c'f are both parallel to the ground line, and each 



THE ELEMENTS OF PROJECTIONS. 



21 




Fig. 5. 



equal to CF. That is, when a line is parallel to the ground line^ 
each of its projections is parallel to the ground line, and equal to 

the line. 

50. Again : AC, for example, is oblique to both planeo of pro- 
jection, but it is in a plane Aa'wc, which is perpendicular to both of 
these planes, and both of its projections, ca and c'a', are perpen- 
dicular to the ground line, and each is less than the line AC. Tha 
is when any line, is oblique to both planes of projection^ and is in 
a plane perpendicular to hoth^ each of its projections is less than 
the line, and is perpendicular to the ground line. 

51. In Fit'". 5, AB is a line which Js paralled to the horizontal 
plane GH, only, ah and a'V 

are its projections. a5=AB, 
and a'V is less than AB and 
parallel to the ground line. 
That is, when a line is parallel 
to the horizontal plane only., its 
horizontal projection is equal 
and parallel to itself and its 
vertical projection is parallel to 
the ground line and less than 
the line itself. 

52. In Fio". 6, AB is a line parallel only to the vertical plane 
YL, ah., its horizontal projection, 
is parallel to GL, and less than 
AB. a'h' , its vertical projection, 
is cqnal and 'parallel to AB. That 
is, ichen a line is parallel to the 
vertical plane only., its vertical 
projection is equal and parallel to . 
the line., and its horizontal projec- 
tion is parallel to the ground line, 
and less than the given line. 

53. Finally, in Fig. 7, AB is a 
line which is oblique to both 
planes of projection. Each of its 
projections, ah and a'h\ is less than AB, and oblique to GL. 
That is, when a line is ohlique to hoth planes of projection, 
hoth of its projections are less than itself and are oblique to the 
ground line. Article 50 is a special case of this principle. 

Klines, in any position, are parallel, their projections will be 
parallel. 




Fig. 6. 



22 



LINEAR T>ERSPECTIVE. 



Lines, like points, are, in the language of projections, named by 
naming their projections (44). Thus, in the three preceding figures, 

the line itself, AB, would 
be designated as the line 
db-ah\ the vertical pro- 
jection being distinguish- 
ed by accents. 

54. The space on the 
same side of the vertical, 
or perspective plane, as 
the eye, is said to be in 
front of it. If, now, the 
horizontal plane, GH, be 
extended back of the ver- 
tical plane CV, Fig. 4, an 
angular space in which objects may be placed, will be formed 
behmd the vertical plane. 





Illustration. — In Fig. 8, let GH be the front part ot the horizon- 
tal plane, and GH' its back part — the vertical plane, GY, being 
viewed in the direction of the arrows. Let KQhd be a square 
prism standing on the back part of the horizontal plane, and with 
its faces parallel and perpendicular to the vertical plane. Then, a8 
shown in the figure, C'D'c'c?', a rectangle, equal to the face CDcc?, 
will be the vertical projection of this prism, and its lower base, 
ahcd^ will constitute its horizontal projection. 

Observe, here, that when a body stands on the horizontal plane, 
its vertical pi'ojcction will stand on the ground line (45). 

Remarks. — a. In practice, and for brevity, the horizontal pro- 



THE ELEMENTS OF PROJECTIONS. 23 

jection is called the plan^ and the vertical projection, the eleva- 
tion. 

h. If, in the three preceding figures, the given lines had been 
placed behind the vertical plane, their vertical projections would 
have remained the same, and their horizontal projections would 
have appeared behind the ground line, and parallel to their present 
positions. The student is recommended to reconstruct these figures 
accordingly. 

55. In (41) projections were spoken of as drawings which show 
objects as they are, rathei than as they appear. That is, projec- 
tions show^ the real forms and dimensions of objects. This will 
now appear from a fuller examination of Figs. 4 and 8. In Fig. 4, 
the prism being placed with its length parallel to the ground line, 
and one of its rectangular faces parallel to the liorizontal plane, its 
horizontal projection gives the true size of that face ; and the ver- 
tical projection, the altitude perpendicular to the face BCEF. 
Hence the prism is thus fully given by its projections, since a body 
is said to be completely given, when, as is true in this case, such 
of its dimensions are known as enable one to find the area of its sur- 
face, or its solidity. 

Still more clearly is it evident, that, in Fig. 8, the plan, ahcd^ 
gives the true width and thickness of the prism, and the elevation, 
the width and height. That is, the two projections, together, give 
the three dimensions of the prism in their real size. 

Thus it will be seen in all the subsequent figures, that the pro- 
jections of objects give their real forms, as seen when looking 
perpendicularly towards the planes of projection, and that when 
these objects are placed in simple positions with respect to those 
})lanes, as they always may be, their projections will give the sim- 
ple dimensions of those objects, as in Fig. 8. 

Note. — The figures in this chapter, and others like them, are nothing else than 
examples of the " military perspective," or oblique projections, explained in my 
" Elementary Projection Drawing," Div. lY., Chap. Y. They are, therefore, as la 
evident by examination, exact constructions, representative of models of construe 
lions in space. 



24 IJNEAR PERSPECTIVE. 



CHAPTER III. 

rUE CONSTRUCTION OF THE PERSPECTIVES OP OBJEOTS FROM THEII 

PROJECTIONS. 

56. The preceding pictorial representatives of models of projec- 
tions, may suffice to render the subject of projections of given 
objects intelligible. We therefore proceed to illustrate the remain- 
ing points in (37) viz. the projections of the point of sight (29), the 
visual rays (30), and the constructions of true perspectives of objects 
from their projections, instead of from the objects themselves (41). 

We have seen that the eye is at an indefimtely great distance 
from each plane of projection, successively, in viewing objects as 
seen in projection (46). But objects, as seen in perspective, are 
supposed to be viewed from points at ordinary finite distances. 

The point of sight is therefore projected like any other point, as 
in Fig. 3. 

57. Since the perspective of a given point in space, is the point 
where the visual ray from that given point pierces the perspective 
plane (35), it is necessary, in the next place, to understand the 
method of finding the point in which a given line pierces any verti- 
cal plane, taken as a perspective plane (36). 

For this purpose, see Fig. 9. Here GHH' is the horizontal plane, 
and GY, the vertical plane. AE is any line in space, joining the 
point A, behind the vertical plane, with the point E in front of it. 
Aa and Ee represent the vertical projecting lines which meet the 
horizontal plane at some points represented by a and e, and which 
therefore determine ae as the horizontal projection of AE. Like- 
wise the projecting lines Aa' and Ee', w^hich are perpendicular to 
the vertical plane, give a'e' as the vertical projection of AE. 

This being established, we have, by referring particularly to the 
line AE itself, the following, as the Jirst method of explaining the 
construction of the desired point. 

It is evident from the figure, as just described, that any line, as 
AE, must pierce the vertical plane somewhere in its own vertical 
projection, ae\ Also, as AE is directly over its own horizontal 
projection, ae^ it must meet the vertical plane in some point directly 



THE CONSTKUCTION OF THE PERSPECTIVE OP OBJECTS. 25 




Fig. 9. 

over the point n, where its horizontal projection meets the vertical 
plane, in the ground line. Hence AE meets the vertical plane at 
n\ the intersection of a'e' with nn', a line perpendicular to the 
ground line at n. 

58. Now suppose the line itself, AE, to be removed, leaving 
Only its projections, ae and a'e\ to be used in finding the point 7i\ 
In this case, we have the following, as the second method — and the 
usual practical one — of explaining the construction of n'. If a 
point lies in the vertical plane, its vertical projection is the point 
itself; and its horizontal projection is in the ground line (45). 
Conversely, if a point in the ground line is the horizontal projec- 
tion of some point, that point is in the vertical plane. Hence in 
Fig. 9, w, where the horizontal projection, ae, of the given line 
meets the ground line, is the horizontal projection of that point of 
the line in which it pierces the vertical plane. This point itself 
being, as already explained, in the vertical projection, a'e', of the 
given line, and also in a perpendicular to the ground line at n, it is 
at n\ the intersection of nn' with a'e\ 

HemarJcs. — a. The construction of n' being of constant occur- 
rence in exact perspective drawings, both of the above explana- 
tions should be memorized, as well as clearly understood, until 
they become thoroughly familiar. 

b. It is now evident that if E is the position of the eye, and 
GLY the perspective" plane, AE is a visual ray, and n' is the per- 
spective of the point A, as seen by the eye at E. 

59. In further illustration of the manner of finding the perspec- 



26 



LINEAR PERSPECTIVE. 



lives of objects from their projections, Fig. 10 is added, which is a 
pictorial representation of the construction of the perspective of a 
straight line in space. 




Fig. 10. 



GHH' is the horizontal plane. GY, the vertical plane, is, as 
usual^ taken also for the perspective plane. Let AB be any- 
oblique line, behind the perspective plane, and meeting the horizon- 
tal plane at B, and whose perspective is to be found. Let E, in 
front of the perspective plane, be the point of sight. By (43) e 
and e' are the projections of the point of sight. Likewise, a' 
and a! are the projections of A. The point B, being in the hori- 
zontal plane, is its own projection on that plane, and by (45) h\ in 
the ground line, is its vertical projection. Therefore, aB and a!h' 
are the projections of the given line. Now, AE is the visual ray 
from A, the upper end of the line AB ; ae and a'e' are the projec- 
tions of this ray. Then by (58) this ray pierces the perspective 
plane at A', which is, therefore, (58&) the perspective of A. Like 
wis-e BE is the visual ray from B, the foot of the given line ; and 
Be and h'e' are its projections. This ray pierces the perspective 
plane at B', which is, therefore, the perspective of B. Hence A'B' 
is the perspective of AB. 

Remarks. — a. Since the perspective, or vertical plane, is placed 
between the eye and the given object, the object lies behind the 
perspective plane, as in Figs. 8 and 10; hence its plan will appear 
behind the ground line. 



THE CONSTRUCTION OF THE PERSPECTIVE OF OBJECTS. 27 

h. In all cases where two planes of projection are used, as just 
shown, the vertical plane of projection is also the perspective plane, 
and, therefore, contains both the vertical projection and the per- 
spective of the given object. 

c. By erasing the lines AB, BE, and AE, so as to leave only their, 
projections, also the projecting lines Aa, Aa\ B^', Ee, and Ee', the 
remaining lines would show pictorially the construction of the per 
spective. A'B', from the projections, only, of AB, and of the ray 
AE. 

The construction of such a figure is left for the student to make. 
[Let each of these examples be worked out twice ; first, as in 
Art. (59), Eig. 10 — that is, hy showing the given lines in space, as 
AB and AE, and the point of sight, Ej second, as indicated in 
Remarh c, above, by shoiuing only the projections, as ah — a'h', of 
these given parts.] 

1. In Art. (59), Fig. 10, let the given line AB be parallel to the vertical 
plane GV (52). 

2. In the same, let AB be parallel only to the horizontal plane GH (51). 

3. Let AB, Fig. 10, be parallel to the ground line GL (49). 

4. Let AB be perpendicular to the vertical plane GV (48). 

5. Construct, on a single figure, as in Fig. 10, the perspectives of several 
lises, all perpendicular to the vertical plane ; some of them in space, as AD, 
and some in the horizontal plane, as ad, Fig. 8, and note that these perspec- 
tives will all pass through the point e'. 



28 



LINEAR PERSPECrnrii;. 



CHAPTER IV. 

REAL PKOJECriONS, AND PERSPECTIVES MADE FfvOM THEM. 

60. All the preceding figures are only the pictures of projeo 
tions, and pictures of the perspectives, made from those projections, 
and not the projections and the perspectives themselves. They are 
pictorial representatives of the models which would show given 
objects, the eye, the perspective plane, and visual rays, as they 
actually exist in space. 

It is, therefore, next to he found how these projections and per- 
spectives themselves are represented. In doing this, we seek first 
the method of representing the planes of projection, which are 
really at right angles to each other, upon a single flat surface, as a 
sheet of paper. 




Fig. 11. 



61. From Fig. 11 it is evident that, if the vertical plane, GW,, 
!)e revolved directly back about the ground fine, GL, as an axis, 
until it coincides with the back portion, GH', of the horizontal 
plane, all the points of the vertical plane w^ill describe circular arcs 
in the direction of the arrows, and in planes perpendicular to the 
ground line. If, then, A be a point in space, and in front of the 
vertical plane, and a and a\ its projections, a\ will describe the 
quadrant a\ a\ and will be found in the line ana\ a perpendicular 
to the ground line through the horizontal projection, a, of the 
given point. 

62. According, now, to this illustration, the following method 



EEAL PROJECTIONS, AND PERSPECTIVES MADE FROM THEM. 29 

is universally agreed upon as the one to be practically ado}. ted in 
representing projections. 



Fig. 12. 

Having drawn any line, GL, Fig. 12, upon the drawing paper, 
lo represent the ground line, it is understood that all that portion 
of the paper below or in front of the ground line, represents the 
front part of the horizontal plane of projection, and that the por- 
tion above or back of the ground line, represents both the back 
part of the horizontal plane, GH', Fig. 11, and the vertical plane, 

Gy;,Fig.ii. 

Hence, make na and na\ Fig. 12, equal respectively to na and 
na\^ or na', on Fig. 11, and a and a' will represent, not the pic- 
tures of the projections of A, but the projections themselves of 
that point. Accordingly, as in (44), the point supposed is named 
by its projections, and we say " the point aa'," meaning the point 
in space whose projections are a and a\ and which is at the height, 
a'n^ above the horizontal plane, and distance, a/i, in fi-ont of the 
vertical plane. 

63. Having now shown, pictorially, the manner of representing 
the projections of points, upon the planes of projection when in 
their real position (43) and as shown after revolution (61) and the 
manner of finding the perspective of a point, or a line (585-59) the 
way is prepared for the connected review embraced in the three 
following figures, which show first : a pictorial representation of 
the construction of the perspective of a point, on the perspective 
plane in its 7^eal position; second: a similar view after the revolu- 
tion of the perspective plane into the plane of the paper ; and 
third: an actual construction of the real perspective of a point. 

64. In Fig. 13, E is the position of the eye, e is its horizontal 
projection, and e' is its vertical projection. Ce' = Ee shows the 



80 



LINEAR PERSPECTIVE. 



height of the eye above the horizontal plane, and C e = E e' shows 
its distance in front of the vertical plane, LGP, which is also the 
perspective plane. 




Fig. 13. 

A is the point, whose perspective is to be found, a is its horizon 
tal projection, and a' its vertical projection. Ba' -= Ka shows 
the height of A above the horizontal plane GHH; aB a'A 
shows its distance behind the perspective plane LGP, and CB 
shows its distance to the right of the observer standing at e. Thus 
the position of the point is completely indicated with respect, 
both to the eye at E, and the perspective plane (22). 
We have, now, the following statement of this problem. 
Given: GHH, 
LGP, 
E, and 
A. 
Required^ the perspective of A, given in its projections a and 
a' ; on LGP, as seen from E, given also by its 
projections e and e'. 
Coneiruction. Draw ae and a'e\ 2ir\(\. note 7i. At n erect 
ns perpendicular to GL and note 5, where it 
meets a'e'. Then 5 will be the required per- 
spective of aa\ that is, of A (58). 
65. This being established, we proceed according to (63) to 
represent, pictorially, the revolution of the perspective plane back- 
ward into the horizontal plane, so as to show the above construc- 
tion in a single plane, as is done in practice. The several quarter 



REAL PROJECrnONS, AND PERSPECTIVES MADE FROM THEM. 31 

circles, Fig. 13, represent the revolutions of the several points a', e\ 
etc., about GL as an axis, till they reach the horizontal plane. 

Thus LGP' Is the revolved position of the perspective plane 
LGP, and a!\ s\ and e" are the revolved positions of a\ s, and e' 
respectively. Then a" e" is the revolved position of the vertical 
projection a' e' of the ray AE, and n s' the revolved position of 
ns, giving s', as the revolved perspective. 

Since the horizontal plane is unmoved, all points upon it remaii 
fixed, and a and a" are the projections of A, e and e" the projec- 
tions of E, and ae and a"e" the projections of the visual ray AE. 

Observe, now, that when the two planes are no longer shown in 
their real position, the ray itself, AE, can no longer be shown, so 
that Fig. 14 shows separately, and pictorially, all of Fig. 13 that 
can appear when the two planes are represented as one surface. 
Fig. 14 evidently shows, however, according to the last article, all 




L 

/H 

^,,-^^^^^^^^^. y ^^ / 

n\ 

^^-" -^^^ \g 

G P 

Fig. 14. 

that is essential in finding s, the perspective of aa' as seen from 
the eye at ee\ 

67. Finally ; Fig. 15 shows the pictorial representation in Fig. 
14, transformed into an actual construction, according to (62). 

[JV^Qte. — While a single complete illustration, fully explained, 
suffices for the purposes of a text book, the learner, in order to 
avoid confusion of mind in his progress, should make himself per- 
fectly familiar with each successive stage of the subject, by con- 
structing a variety of figures, similar to those thus far given.] 

Proceeding, now, with Fig. 15, aa' is the given point, at a distance 
equal to ad, back of the vertical plane, and at a distance equal to 
a'd above the horizontal plane, ee' is the point of sight, at a dis- 
tance equal to eb in front of the vertical plane, and at a distance 
equal to e'5, above the horizontal plane. 

From the preceding descriptions, it follows that ae is the horizon- 
tal projection of % visual ray from the given point aa\ and a'e' is 



32 



LINEAR PERSPECTIVE. 



its vertical projection. Then p^ the point where the horizontal 
projection of this ray meets the ground line GL, is the horizont;il 
projection of that point of the ray in which it pierces the vertical, 




Fig. 15. 



I. e. the perspective plane (58). The latter point being also in the 
vertical projection of the ray (57,58) is at P, the intersection of a'e' 
with ^P, a perpendicular to the ground line at p. Therefore P ia 
the perspective of aa', as seen from the point ee' . 

68. In order to find the perspective of any object, we have only 
to find the perspectives of its separate points, exactly as just 
described. Hence the following explanations will not be so minute 
for each point, as the one just given. We shall now proceed to 
explain the construction of the perspectives of the leading elemen- 
tary solids, viz. the prism^ pyramid^ cylinder^ cone^ and sphere ^ toge- 
ther with various subordinate practical particulars. And though this 
may not in itself mtQve&t the learner as much as the representation 
of objects whose perspectives have more of pictorial effect, yet the 
recollection that no other method will so concisely afford an equally 
abundant variety o^ universally useful methods of practical ope- 
ration^ in subsequent practical examples, may sufiice to com- 
pensate for the comparative inelegance of the perspectives now to 
be explained. 

Example 1. — To find the Perspective of a Vertical Square 
Prism, situated as shown in Fig. 8. 

From Fig. 8 it is evident, that when the perspective plane is 
revolved backward, the lines c'C and d'T)' of the elevation (55a) will 
exactly fall upon the lines ch and da of the plan. Hence, in Fig. 16, 
ABCD and c'd' CD' are the correct projections of a square prism 



KEAL PEOJECTIONS, AND PERSPECTIVES MADE FROM THEM. 33 

Standing on the horizontal plane, at the distance d'T> behind the 
vertical or perspective plane. 







'^*^! 
■-^ 



Fig. 16. 



Let EE' be the position of the eye. The rays from c' and C, 
being one directly over the other, have the same horizontal projec- 
tion, CE. Then by (67) CE — c'E' is the visual ray (58^>) from the 
front left hand corner, Cc', of the base, and/* is the perspective 
of that corner. CE — C'E' is the ray from the corresponding upper 
corner CC. It pierces the perspective plane at F, which is there- 
fore the perspective of CC Likewise the visual ray DE — c?'E' 
pierces the perspective plane at o, which is therefore the perspec- 
tive of the point Dd' ; and the visual ray DE — D'E' pierces the 
perspective plane at O, giving the perspective of the point DD'. 
Hence the figure F/Oo is the perspective of CD — c'd' CD', the 
front face of the given prism. Finally, BE and c?'E' are the projec- 
tions of the visual ray from the right hand back corner, B, d\ of the 
base of the prism. This ray pierces the perspective plane at n, the 
perspective of this corner. Also BE — D'E' is the visual ray from 
the corresponding corner, B,D', of the upper base. This ray gives 
N as the perspective of B,D'. Then drawing ON", on, and 9iN, we 
shall have the complete perspective of the visible edges of the 
given prism. 

69. JRemarks, — a. Observe that/b and FO are, by the construc- 
tion, parallel to the ground line GL, and to the lines CD — c'd\ and 

3 



34 



LINEAR PERSPECTIVE. 



CD -CD', of which they are the perspectives. Also that the right 
hand edges, BD — d' and BD — D', of the bases, are perpendicular 
to the perspective plane, while their perspectives on and ON meet 
at E', if produced. This is easily explained from elementary geo- 
metry. Planes containing the eye and the vertical edges, as at C 
and D, are vertical planes, standing on the lines CE and DE. 
Hence they must intersect the perspective plane, which is also ver- 
tical, in lines, A;F and AO, which will be vertical, that is parallel to 
the vertical edges of the prism, at C and D. 

In like manner, since the top and bottom edges, dd' and CD', 
are parallel to the perspective plane, the planes passed through 
them and the eye, must intersect the perspective plane, in lines /b 
and FO, which will be parallel to those given edges. And, gen- 
erally, the perspectives of all parallels to the perspective plane, are 
parallel to the lines themselves. 

h. Planes through the eye, and the edges DB-J' and DB-D' 




> X ^ V^Ky 



'» N\ 



Fig. 17. 






n 



which are perpendicular to the perspective plane, will contain tne 
visual rays from D,c?' and DD' and will, themselves, be perpendicu. 



REAL PROJECTIONS, AND PERSPECTITES MADE FROM THEM. 3& 

lar to that plane. They will therefore intersect each other in a line, 
perpendicular to the perspective plane at E'. Hence their inter- 
sections with the perspective plane, which will be the indefinite 
perspectives (37) of the edges contained in them, will be the lines 
d':E\ and D'E'. 

Example 2. — To find the Perspective of a Triangular 
Pyramid, 

This example embraces the perspectives of oblique lines, abc — v 
and a'b'c' — v\ Fig. 17, are the projections of a triangular pyramid^ 
standing on the horizontal jjlane, and behind the vertical plane. E 
and E' are the projections of the point of sight. oE — a'E' is the 
visual ray from the point aa' of the base of the pyramid. This 
ray pierces the perspective plane at A, the intersection of /"A and 
a'E'. A is therefore the perspective of aa'. Likewise, by the rays 
bE — b'E\ and cE — c'E', we find B and C, the perspectives ofbb' 
and cc\ wE — -y'E' is the visual ray from the vertex vv\ and it 
pierces the perspective plane at Y, the intersection of hY and 
v'E', giving V as the perspective of vv'. 

Joining the points now found, ABC — V is the perspective of 
the given pyramid abc-v — a'b'o' — v\ as seen from EE', 

Examples for Practice. 

1. In Fig. 16, let the prism be a wire skeleton, so that all the edges will be 
visible in perspective. Then find the perspectives of all those edges. 

2. In Fig. 16, let EE' be to the left of CA, and E' above Q'\i\ and then 
find the perspective of the prism. 

3. In Fig. 17, let the pyramid have a square base, with one side parallel to 
the ground line. 

4. General Example. — In either of Figs. 16, 17, let the base be either a 
hexagon or an octagon ; and the body either solid, or a wire skeleton show- 
ing every edge. 

Note. — Though parts of the next two chapters may seem dry and uninteresting, 
owing to lack of pictorial effect or of obvious immediate usefulness, yet they 
should be thoroughly learned, with interested attention, because they concisely 
explain the principles which, when mastered, will enable the learner to correctly 
make the perspectives of architectural exteriors and interiors, even when, quite 
elaborate, by the very simple and convenient methods there explained. 



dti 



LINEAR PERSPECTIVE. 



\ 



CHAPTER Y. 

BEMOVAL OF PRACTICAL DIFFICULTIES ARISING FROM THE CONFU 
SIGN OF PROJECTIONS AND PERSPECTIVES. 

§ I.-— 'First Method, Translation, forward, of the Perspective 

Plane. 

70. The perspective plane being between the eye and the given 
object, the plan of that object must lie behind the ground line. 
Also, as the perspective plane contains both the vertical projection 
and the perspective of the object, these two must both fall upon the 
plan, when the perspective plane is revolved back into the horizon- 
tal plane ; as seen in the last two examples. 

The confusion of lines arising from this source is sufficiently ap- 
parent from Figs. 16 and 17, though they embrace very simple 
objects, and remove the perspectives as far as possible from the 
projections, by placing the eye considerably to one side of the 
projections. 

71. Hence, before proceeding further with practical constructions, 
we shall present a simple method of obviating the difficulty just 
mentioned. This method consists in transferring the perspective 
plane, with all the points in it, directly forward, far enough to allow 
it to be revolved back so as to lodge the figures in it entirely below, 
or in front of, the plan. 




Fig. 18. 

This method is illustrated in Fig. 18. A is a point whose projec 
tions are a and a\ on planes seen edgewise and in their real posi 
tions at riirht ansjles to each other, at aGG' and GP. E is the 
place of the eye. Then X represents the perspective of aa'. 



REMOVAL OF PEACTICAL DIFFICULTIES. 37 

When, now, the perspective plane GP is revolved back as shown 
by the arrows, carrying a' and X to a'^ and X', a,X' and a" will be 
crowded together. But suppose the perspective plane to be first 
moved forward — carrying along the points a' and X — to a new 
position G'P', and then to be revolved. The perspective, X'', will 
then appear at X''^, free from the plan ; and it may also be freed 
from the elevation, in practice, by erasing portions of the latter 
from time to time, as the construction of the perspective progresses, 
or by transferring only the perspective points. 

The elementary examples of the last chapter are here continued, 
according to the method just explained. 

Example 3. — To find the Perspective of a Cube, "which 
stands obliquely ^with respect to the perspective plane. 

See Fig. 19. aceg is the plan of a cube thus situated, and a'b'c'f^ 
is its elevation. 

The ground line GL indicates the first position of the perspective 
plane, and G'L' shows its position after translation forward. E is 
the horizontal projection of the point of sight. Being in the hori- 
zontal plane, its position is not afiected by the translation of the 
perspective plane. E' is the vertical projection of the point of 
sight, shown only on the second position of the perspective plane, 
since it is used only there. For a similar reason, the vertical pro- 
jections of the visual rays are shown only on the second position 
of the perspective plane. aE is the horizontal projection of the 
visual rays from the two points aa' and a,6' (Ex. J.). By making 
h''a''^=iVa\ and in a'V produced, we find the projections of a' and 
y upon the second position of the perspective plane. Likewise 
we find/'', e'', c", etc. Then, for example, «E and a"W are the 
projections, employed, of the visual ray from a^a" \ or, more 
briefly (58) aE — a^'W is the visual ray from a,a". This ray 
pierces the perspective plane at A, the intersection of a"E' with 
the perpendicular to GL, at A, where the horizontal projection, 
aE, of the ray meets the real^ that is the original position of the 
ground line (57-8). Then A is the required perspective of a^a'\ 
Other points as B,F, etc., of the perspective of the cube may be 
found in a precisely similar manner. The construction of some of 
the points is therefore omitted, to avoid unnecessary confusion of 
the figure. Thus, the perspective of the point, c^d" will be at the 
intersection of a line c?"E' with the perpendicular to GL at n. The 
perspective of the back upper corner g^g" is likewise at the inter- 



38 



LLN^EAR PERSPECTIVE. 




Bection of a ray from g^' to E\ with the perpendicular to GL 

at o. 

To avoid the acute intersections, as at B, by the method of two 

planes, without setting E,E' far to one side, as in Fig. 17, trans- 
late the points, as 
a,6", of the given 
object, only, to one 
side in a direction 
parallel to the 
ground line, and 
then find their per- 
spectives, as B' (not 
shown) which will 
be well defined. 
Then a parallel to 
the ground line, 
through B', will in- 
tersect either 6''E', 
or AB, giving B by 
a well defined in- 
tersection. Observe 
that E,E' is not 
moved. 

Remark. — The 
perspectives of other 
plane-sided objects, 
in various positions, 
should be construct- 
ed by the learner, 
by the method just 
explained. For ex- 
ample, let Fig. 17 
be re-constructed 
according to the 
method of Fig. 19. 







Ql 



Z^/ 



• 5'" •' 



Fig. 19. 



§ II. Second Method. Use of three Planes, 



72. The confusion of the diagrams, arising from the confounding 
together of the perspective with either or both of the projections 



REMOVAL OF PRACTICAL DIFFICULTIES. 



36 



of the given object, may be still further avoided by making the 
perspective plane a third plane, separate from both of the planes 
of projection, and at right angles to both of them. 
This is accomplished in the manner illustrated in Fig. 20. 




Fig. 20. 

OHH is the horizontal plane of projection ; VV, the vertical 
plane of projection, and OLQ the perspective plane. P is a point 
in space, whose perspective is to be found, p represents its hori- 
zontal, and jt?' its vertical projection. E is the position of the eye, 
e its horizontal, and e' its vertical projection. Then PE represents 
the visual ray, whose intersection with OLQ will be the perspec- 
tive of P. pe is the horizontal, and p' e' the vertical projection 
of this ray. The perspective plane OLQ is perpendicular to both 
of the other given planes, and LQ is its intersection with the ver- 
tical plane of projection. LQ is called the trace of OLQ upon the 
vertical plane of projection. Then, as in previous cases, Pi, the 
perspective of P, is in the line n Pj, perpendicular to the ground 
line OL at n. Likewise it is obviously in the line r Pj, perpendicu- 
lar to the trace LQ at r. Hence P is at the intersection of n Pi 
and r Pi. 

73. Now in order to bring all three of these planes into a single 
surface, as is done in practical drawing, the perspective plane may 
be revolved about its trace LQ till it coincides with the vertical 
plane W, which may then be revolved back as usual around the 
principal ground line, HLj. But by such a proceeding, the per- 
spective of an object would by revolution fall upon the vertical 



40 



LINEAR PERSPECTIVE. 



projection of that object. Hence the perspective plane is moved 
towards the eye, and parallel to its first position to some con- 
veuient new position as nJUiT^, before being revolved. Then, as 
every point of the perspective plane moves parallel to the ground 
line, n will appear at ni, and r at r^, and after revolution in the di- 
rection niTij, the vertical line nP^ will appear at n^Fz^ and th 
horizontal line rP,, at rj?2. Hence Pa will be the perspective of 
P, after the translation and first revolution of the perspective 
plane. 

14. The perspective of a point by the method of three planes, 



T% 



J,....^. 



r 



,\n 



-^' 



JJLL 



,.*e 



:.--..-. mt 



Tj. 



Fig. 21. 



shown plctorially in Fig. 20, is shown as an actual construction in 
Fig. 21. The former figure is exactly transformed into the latter 
by making the corresponding distances equal in both, and by letter- 
ing the same points with the same letters, so far as shown at all. 

pp' is the given point, given by its projections, ee' is likewise 
the point of sight, nLr the first, and nJ^iTi the second position of 
the perspective plane, thus indicated as at right angles to both 
planes of projection, pe — p'e' is the visual ray from pp\ which 
pierces the perspective plane riLr at a point whose projections are 
n and r. After translating this plane, parallel to the ground line, 
to the position njitlr^^ these points appear at n^ and 7\. Then, by 
revolving the perspective plane from njjiri into the vertical plane 
of projection, the point Uir^ describes a horizontal arc about the 
point Li, r^ as a centre. The projections of this arc are Jiin^ and 
r-iP , and Pa thus appears as the perspective of pp'. 



REMOVAL OF PRACTICAL DIFFICULTIES. 



41 



Remarks. — a. The perspective plane must, in Fig. 21, be trans- 
.ated to the right so as to revolve to the left, in order that the right 
Hand of the perspective may continue to correspond with the right 
hand of the object drawn. This will be obvious on inspection in 
the succeeiing examples, wherever three planes shall be used. 

b. Either of the methods of disposing of the pei-spective plane, 
explained in this chapter, will be used at pleasure in the solutions 
which follow. The student is advised to solve the subsequent 
problems, on three planes, when two are used by the author, and 
vice versa. 

To assist therefore in becoming more familiar with the use of 
three planes, the following practical problem is given. 



V'yr^., 




Fia. 22. 



Example 4. — To find the Perspective of an Obelisk, com- 
posed of a frustum of a lon^ pyramid, cappea oy a snorx 
pyramid. 



42 LINEAR PERSPECTIVE. 

Let the square acbd — c'd\ Fig. 22, be the horizontal and verti- 
cal projections of the base of the obelisk ; and v-jnot — v'-n'f the 
projections of the cap pyramid. 

Let PQP' be the first and real position of the perspective plane, 
at right angles to bo*.h planes of projection. Let PiQiP'i, be its 
second position, parallel to the first, from which it is revolved 
around P'lQi, its intersection with the vertical plane, until it coin 
cides with that plane. EE' is the point of sight. 

To find the perspective of any point, as aa\ of the base. aE and 
a'E' are the projections of the visual ray from this point. This ray 
pierces the perspective plane at gg'. This point, after translation, 
appears at g^ ^/,. found by drawing gg^^ and g'gi\ parallel to the 
ground line. After its revolution thiough the horizontal quarter 
circle whose projections are ^1^2 and gi A, it appears at A, the inter- 
section of ^I'A with giA, perpendicular to the ground line QQi. 

In like manner C and B, the perspectives of cc' and bb' are 
found. 

JVbte that bb', the invisible corner of the base as seen in ver- 
tical projection, is the right hand corner, to the eye at EE' looking 
in the direction E?;. 

To find the perspective of any point of the cap pyramid, we also 
proceed just as before. Thus, oE — o'E' is the visual ray from the 
corner 00'. This ray j)ierces the perspective plane PQP' at pp\ 
which is translated to pip/, and from that position revolved in a 
horizontal arc, as before, to O, the perspective of 00'. 

HemarJcs. — a. Every point of the perspective being thus found 
in precisely the same manner, the construction of several of them 
is left to be made by the student. 

b. Observe also, that as the operations in Figs. 21 and 22 are 
precisely similar, the perspective of any object, by the method of 
three planes, is simply, and only, a continued repetition of the con. 
gtruction of the perspective of a single point, as in Fig. 21. 

c. Practice is required, however, to enable the learner to under 
stand readily the form and position of any given object from it$ 
vrojectio7is, and to determine easily, by mere inspection, the pro- 
jections of those points which are seen from the given point of 
sight. Hence, again, the student is advised to construct the per- 
spectives of various other simple objects, from their projections, as 
in this example. 



PliOJECriONS AND PERSPECTIVES OF CIRCITLAR BODIES. 43 



CHAPTER VI. 

PROJECTIONS AND PERSPECTIVES OF CIRCLES, AND OF BODIES HAVING 
PARTLY OR WHOLLY CIRCULAR BOUNDARIES. 

75. The outlines of almost all artificial objects will be found, by 
analyzing them, to consist of straight lines and circular lines. 
Having now shown how to find the perspectives of points, straight 
lines, and plane sided figures, both pictorially and by actual con- 
struction, we next proceed to explain the construction of the per- 
spectives of cii'cles, and of various bodies bounded in part, at least, 
by circles. 

Example 5. — To find the Perspective of a Circle lying in 
the horizontal plane. 

The method by two planes, with the vertical or perspective plane 
translated forward before being revolved back into the horizontal 
plane (71) is here employed. See Fig. 23. 

Let OAide be the horizontal projection of the given circle. As 
this circle lies in the horizontal plane, its vertical projection, a'd\ 
must lie in the ground line LL (45). Now let the perspective 
plane, which is perpendicular to the paper at LL, be translated 
forward to the parallel position L'L', and then, as usual, revolved 
backwards into the horizontal plane, or plane of the paper. Then 
take EE' as the point of sight, and let all the vertical projections 
be shown on the translated position of the perspective plane. 
Accordingly, a"d" will be the new vertical projection of the given 
circle. 5E is the horizontal, and/*"E' the vertical projection of the 
visual ray from the point h^f in the circle. The point ?^, where 
the horizontal projection JE meets the ground line, LL, is the 
horizontal projection of that point of the ray itself in which it 
pierces the perspective plane (58). The latter point is at once in 
the perpendicular, wB, to the ground line, and in the vertical pro- 
jection f'Yl of the same ray. Hence the desired point is B, 
which is the perspective of J, /''. 

[This being a new form of example, the construction of the per- 
spective of one point is explained as minutely as if it had not been 



44 



LINEAR PEESPEOriVE. 



fully explained already. The details of the explanation wiH there 
fore be omitted in future similar constructions.] 




Fig. 23. 

The ray cE-c"E' pierces the perspective plane at C, which is 
therefore the perspective of c^d' , In like manner the pei*spectives 
of any other points can be found. 

By inspection of the vertical projection, a"d\ it appears that 
the extreme visual rays, as a"E', as seen in vertical projection, are 
those which proceed from the opposite ends of the diameter whose 
horizontal projection is at. Hence rays from points, as g^ before 
that diameter, or J, behind it, find their vertical projections^ ^"E 
andy'E' within a"E'. Hence no point of the perspective of the 



I 



PEOJECTIOXS AND PERSPECTIVES OF CIECULAE BODIES. 



45 



circle can appear outside of the ray a^E', and therefore the per- 
spective must be tangent to a"E', at A, the perspective of aa" , 

The similar result, at the persj3ective of ?, is not shown, as it 
could not appear distinctly on account of the position of EE'. 

The rays whose horizontal projections are tangent to the plan 
at g and (f, include the other rays between them. Hence all points 
of the perspective are betw^een the perpendicalais mD and AC, and 
tlie perspective is tangent to these perpendiculars at D and G. 

The perspectives of tangents, parallel to the ground line, will 
be tangents to the perspective and parallel to L'L'. Having now 
six tangents with their points of contact, besides other points, the 
perspective curve can be very accurately sketched. 

'76. In the previous perspectives ot plane-sided figures, which 
are distinguished by well defined edges and corners, however viewed, 
it will be observed that it can be determined, by simple inspection, 
which edges will be visible from the point of sight. But, in the 
case of objects bounded partly or wholly by continuous curved 
surfices, the consequent partial or total absence of limiting edges 
makes it necessary to discover the visible boundaries by more or 
less of preliminary construction. Hence, a few additional de6ni- 
tions and principles are introduced here for use in the follow^ing 
problems : 

77. Other planes than the planes of projection, go by the general 
name of auxiliary planes. 

Their positions are indicated by their intersections with the 
planes of projections, called their traces. 

Each of these traces takes its name from the plane of projection 
in which it is found. 

78. The point Avhere either trace meets the ground line ia 
where the plane cuts the ground 

line; hence hoth traces of a plane 
must meet the ground line at the 
same pointy if they meet it at all. 
The traces of a plane will meet 
the ground line unless the plane 
is parallel to that line. 

79. If, as ill Fig. 24, a plane is 
vertical, but oblique to the ver- 
tical plane of projection, its verti- 
cal trace, YT, will be perpendi- 
cular to the ground line, G L. 

80. If, as in Fig. 25, a plane is perpendicular to the vertical 



* 




Fig. 24. 



46 



LINEAR PERSPECTIVE. 



plane of projection, its horizontal trace, HT, will be perpendicular 
to the ground line, G L. 

If a plane is perpendicular to both of the planes of projection, 
both of its traces will be perpendicular to the ground line, as we 
have seen in (72-74). 




Fig. 25. 



81. Again ; when a plane is vertical, that is, perpendicular to 
the horizontal plane, all points and lines in it are horizontally pro- 
jected in its horizontal trace ; since the horizontal projections of 
points and lines are vertically under the points and lines themselves. 

Likewise, when a plane is perpendicular to the vertical plane of 
projection, all points and lines in it find their vertical projections in 
its vertical trace. 

82. Any plane containing the point of sight, contains an indefi- 
nite number of visual rays, whose directions radiate in all direc- 
tions, in that plane, and from the eye. Hence such a plane is called 
a visual plane. 

83. A visual plane being thus composed of visual rays, if such 
a plane be passed through a line whose perspective is to be found, 
the trace of that "visual plane on the perspective plane will be the 
perspective of the given line. See Ex. 1, Rem. 5, also Fig. 10, 
where the plane triangle EAB serves to mark the visual plane of 
indefinite extent, and containing the line AB. A B, is a portion of 
the trace of this plane on the perspective plane, and is, therefore, 
the perspective of AB. 

84. For the reason just given (82), the pomt or line at which a 
visual plane is tangent to a curved surface, is a point or line of the 



PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES. 



47 



visible contour of that surface. The perspective of this visible 
contour or boundary, is the boundary of the required perspec- 
tive. (32.) 

Ex. 6.— To find the Perspective of a Cylinder, standing 
on the horizontal plane. A cylinder, seen from above, as it 
stands on the horizontal plane, appears only as a circle. As seen 
ooking forward at it, perpendicularly to the vertical plane, its 
diameter and height are visible. Hence the circle afm, Fig. 26, 
and the rectangle n'o'pr' are the projections of a cylinder in the 
given position. 



JTL 




a-' C £' Jt 



Fig. 26. 



Sj.; 



1& 



This being established, EA, tangent to the horizontal projection 
of the cylinder, is the horizontal trace of a vertical visual plane^ 
tangent to the cylinder along a vertical line of its convex surface at 



48 LINEAR PEKSPjSCnVE. 

h. Likewise, Ea is the horizontal trace of a similar plane, tangent 
to the cylinder along a vertical line at a. The A^ertical projections 
of these lines are h'k' and a'h\ and they are the projections of the 
visible boundaries of the convex surface, as seen from EE'. The 
tangent planes being vertical, their horizontal traces, as EA, are 
the horizontal projections of both of the visual rays, as EA-E'A' and 
EA ¥Jk\ from the lower and upper extremities of the lines of con 
tact, as h-h'h'. (81.) 

This being understood, nothing peculiar remains in the con 
struction of the perspective, ABFK, of the cylinder. Thus, the 
perspective of the point aa' of the lower base, is found by drawing, 
the visual ray, aE-a'E', which pierces the perspective plane at A, 
the intersection of a'E' and g'A, perpendicular to GL at q. Like 
wise, K, the perspective of the point A, h' of the upper base, is ai 
the intersection of h'YJ and sK, ^'E' being the vertical projectior 
of the visual ray from A, h\ and sK. the perpendicular to GL fron' 
the intersection of GL with AE, the horizontal projection of the 
same ray. 

The perspective bases are tangent, as at A and B, to the extreme 
visible elements, as AB ; for the visual plane containing such ex- 
treme element, as a — a'h' is tangent to the visual cone from either 
base. Therefore, the intersections, as AB, of the visual plane, and 
AifF, or BK, of the visual cone, with the perspective plane are 
tangent to each other, as at A. (See Art. 85.) 

MemarJcs. — a. Since there will, even when great care is taken, 
often be slight instrumental errors in the construction of points, the 
curves in the perspective can be more advantageously drawn by 
carefully connecting a few carefully constructed points by easy 
curves, than by finding many points in those curves." 

h. The figures in this book being designed for purposes of instruc- 
tion, necessarily show Ihe lines of construction much more fully than 
is necessary in practice. For example, in finding the point B, Fig. 
26, it is not necessary actually to draw either aE, J'E', or g'B, but 
only to mark the point q in the line aE, then to draw little frag- 
ments of 6'E' and g'B, just- at their intersection B. 

Likewise in Fig. 22, all that is essential in finding O, for example, 
after drawing oE — o'E', is to make ^I'O equal to Qjt?, and in the 
horizontal X\\\<dp'p^. 

c. Another matter of still greater pi-actical importance, is the 
order in which the lines of construction should be drawn. All 
the lines necessary for finding the perspective of one pointy should 
he drawn, before proceeding to draio those by ivhich a new 



PROJECTIONS AND TERSPECTIVES OF CIRCULAR BODIES. 



49 



point isfoimd. Thus in Fig. 26, draw «E, a'E', and ^'A, which give 
the point A. That step being finished, proceed to draw cE — c'E' 
and It which determine t y &c. 

If, on the other hand, all the lines to E', for example, be drawn 
before drawing any to E, there will be a considerable liability to 
mistake in noting wrong intersections ; which could not possibly 
lappen by the first method of operating. 

d. In every case like this, where points of the vertical projection 
are shown only on the second position of the perspective plane, its 
first position, at GL, need not be suj^posed to be revolved back into 
the horizontal plane, but to remain perpendicular to the paper till 
after translation to its second position at GX'. 

85. The convex surface of a cone, as V-ATB, Fig. 27, is com- 
posed of straight lines, which meet at its vertex, V. Hence a plane 
may be made to rest on this convex surface, along any one of its 
straight lines. This plane will be tangent to the convex surface. 

When this plane contains the point of sight from which the cone 
is viewed, it will be a visual tangent plane, and the line on the cone 
along which it is tangent, will be a boundary between the visible 
and invisible portions of the convex surface of the cone. There 
will evidently be two such tangent visual planes, and boundaries, 
for any cone. 




Fig. 27. 

In Fig. 27, let E be the point of sight from which the cone 
V-ATB is seen. Then, as all the lines of the convex surface meet 
St the vertex, V, the two tangent visual planes through E, will alsc 



50 LINEAR PERSPECTIVE. 

pass through V, and hence both will contain the visual ray EV 
through the vertex. 

Now let HK be a horizontal plane on which the cone V-ATB 
stands, and let P be the point where the visual ray EV pierces this 
pJane. 

It will then be evident on inspection of this figure, or of a pape, 
model such as the student can make ; 1° : That as EP is a line com 
mon to both of the tangent planes, P, where EP meets the plane 
HK, will be a point common to the traces (77) of both of the tan- 
gent planes upon the plane HK. 2° : That these traces, being the 
intersections of planes with each other, will be straight lines ; and 
3° : That as each visual plane is tangent to the cone along a straight 
line of its convex surface, the trace of either visual plane upon HK 
will be tangent to the base of the cone, which lies in the plane HK. 

Therefore to find T, and hence TV, a line of contact of a tangent 
visual plane with the cone, draw PT, tangent to the cone's base, 
and TV will be the cone's apparent contour on one side, as seen 
from E. Likewise draw P^, and ^V will be the opposite visible 
boundary of the cone's convex surface, seen from E. 

We will now proceed to show this pictorial illustration in true 
projection, with the perspective of the cone. 

Example 7. — To find the Perspective of a Cone, standing 
on the horizontal plane. 

1.° — Preliminary explanation of the projections. Fig. 28. 
Three planes are here used, the horizontal and vertical planes of 
projection ; and the perspective plane, placed at right angles to 
both of them. 

The cone is supposed to stand on the horizontal plane. In this 
position, its horizontal projection is a circle V-TAB, and its verti- 
cal projection is an isosceles triangle V'A'B', whose base equals the 
diameter of the base of the cone. 

2°. — Construction of the apparent contour of the cone. Let E 
and E' be the projections of the point of sight. Then VE and V'E' 
are the projections of the visual ray through the cone's vertex. 
This ray is common to the two tangent visual planes which contain 
the visible limits of the cone's convex surface. Now to find N, 
where this ray pierces the horizontal plane. If a point is in the 
horizontal plane its vertical projection will be in the ground line, 
hence, conversely, that point, as N', of the vertical projection of a 
line, which is in the ground Ihie, is the vertical projection of that 
point, as N, which is in the horizontal plane (45). Hence the ray VE- 
V'E' pierces the horizontal plane at N, and by (85) NT and N^ art 



PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES. 



51 



the horizontal traces of tangent visual planes, and TV-T'V' and 
tV-t'V are their elements of contact with the eiven cone. These 




Fig. 28. 

elements, with the portion TA^ of the base, form the cone's appa- 
rent contour. 

3°. — JVow to find the perspective of this contour^ that is, of the 
cone. The preceding topics (1° and 2°) contain all that is peculiar to 
this problem. The construction of the perspective is the same as in 
previous examples where three planes were used. The construc- 
tion of \\ only, is therefore explained to assist in the outset of the 
solution, and the rest of the figure is left to be traced out by the 
student. The visual ray,' VE-V'E', pierces the perspective plane 
in its real position, PQP', at aa'. After translating this plane to 
the right, to any convenient distance, as at PfQiP/, it is revolved 
about its vertical trace QiP/, as an axis, and into the vertical plane. 
Thus aa' proceeds to a^a^ and then revolves in the arc a^a.-ajy 
to V", the perspective of VV. (74.) 

By constructing, in the same way, the perspectives of TT' ; AA', 
the point of the base ^hich is rearest to the eye ; and tt\ the per- 
speetive figure can be completed. 

Remark. — ^The eye being here placed below the vertex, N and E 
fall on the same side of the cone, less than half of which is therefore 



52 



LINEAR PEESPECTIVE. 



visible. When, as in Fig. 27, the eye is above the vertex, P (coirc>- 
spending to N in Fig. 28) and E, are on opposite sides of the vertex, 
and evidently more than half of the cone will be visible. Let the 
student construct the perspective of a cone under the latter condi- 
tion, as above, or taking the vertical plane as the perspective plane, 
as }n Fig. 26. 

86. Of the five chief solids of elementary geometry (68), the 
cone is the one which embraces in its outline the three primary 
geometrical elements, viz. the pointy straight line, and circle. 
Hence the most instructive variety of operations will be found in 
constructing perspectives of cones in various positions, such as the 
following. 

Example 8. — To find the Perspective of a right Cone 
with a circular base, -whose axis is parallel to the ground 
line. 




Ai\i\ i\ h 




h' 



Fig. 29. 

We shall employ two planes, and will first explain the projections 
of the cone. Let GL, Fig. 29, be the first, and G'L' the second 



PKOJECTIOXS AND PERSPECTIVES OF CIRCULAR BODIES. 53 

position of the perspective plane, and let the vertical projection be 
shown on this second position only. 

When the axis of a cone, of the kind here given, is parallel to 
the ground line, it is parallel to both planes of projection, and tbe 
projections of the cone will be simply two equal isosceles triangles^ 
with their bases perpendicular to the ground line. Thus YCD is 
the horizontal projection of the given cone, and V'A'B' is its vcr 
tical projection. 

According to (44) A' is the lowest, and B' the highest point on 
the cone's base. In looking dow^n on a cone, these points will appear, 
one directly under the other and in the middle of the width of the 
cone. Hence on the plan, YCD, the point B is the horizontal pro- 
jection of both A' and B'. • 

In like manner (44) C is the foremost and D the hindmost point 
on the cone's base, and C, the middle point of the height of that 
base, is the vertical projection of both of these points. 

To find the projections of any other point in the base. Assume 
fas the horizontal projection of two such points, since a vertical 
chord will contain two points, one over the other, of a vertical cir- 
cle, and two such points will appear in plan as one point. The ver- 
tical projection/", of/", cannot be immediately found, since the line 
from/* to/" coincides with A'B' and hence finds nothing to inter- 
sect at f\ which must therefore be found in some other way ; for 
it is a law of all constructions in drawing^ that a point is always 
found as the intersection of two knov^n lines. 

Accordingly, revolve the front semicircle, BC, of the base to the 
position BC", parallel to the vertical plane. The points y*will then 
appeal- at/"". The vertical projection of the semicircle after revo- 
lution, is the semicircle By""A', and the points/" will be vertically 
projected at /'" and g'". By revolving the semicircle back to its 
true position, remembering that the axis of revolution is the verti- 
cal diameter B-B'A', the points/'/"" and/"' g'" will revolve back 
in the horizontal arcs whose projections are f'f-f'f and f'f- 
\)"'g\ giving g' and f as the desired vertical ]3rojections of the 
points whose plan is f. 

The projections of the cone being now fully explained, nothing 
peculiar to this problem remains ; the construction of the perspec- 
tive being the same familiar operation already often repeated. Thus, 
EE' being the point of sight, VE-VE' is the visual ray through 
the vertex which by (58) pierces the perspective plane at v^ which 
is therefore the perspective of W. E/is the horizontal projection 
of both of the rays whose vertical projections are ^'E' and/"E' 



54 LINEAR PERSPECTIVE. 

These rays pierce the perspective plane at H and F, the perspectives 
o^fy' and^'. Finding other points of the base in like manner, 
and joining them, will give the perspective of the base. The per- 
spective of the convex surface of the cone consists merely of two 
lines from v, tangent to FHc?, the perspective of the base. 

Remarks. — a. The construction of the perspective of the base is 
the same that would be used in finding the perspective of any verti 
cal circle which should be also perpendicular to the vertical plane. 

h. The same construction of f and g' would be required in the 

use of three planes of projection. The perspective of such a circle 

by the method of three planes is left as an exercise for the student. 

Example 9. — To find the Perspective of a Cone, "whose 

axis is parallel to the vertical plane only. 

All that is peculiar to this, and the following example, being the 
construction of the projections of the cone, they only will be 
explained ; leaving the construction of the perspectives as an exer- 
cise for the student. 

By (52) it is evident that VO and V'C, Fig. 30, may be taken 
as the projections of the axis of a cone having the given position, 
VO being parallel to the ground line GL. The axis being parallel 
to the vertical plane, the base of the cone will be perpendicular to 
the same plane and A'B', perpendicular to V'C, and bisected at C, 
will be its vertical projection. 

Four points of the horizontal projection of this base are readily 
found. A' and B', the highest and lowest points, are horizontally 
projected at A and B, on the line ABV, which is the common hori- 
zontal projection of the axis, and of that diameter of the base, which 
is parallel to the vertical plane. C, the vertical projection of the 
foremost and hindmost points, is horizontally projected at C and D ; 
by making 00 = OD = A'C. 

A circle, seen obliquely, appears as an ellipse. Accordingly an 
ellipse, or a smooth oval curve representing one, may now be traced 
through the points A,C,B, and D, by the aid of an irregular curve 
(9). Four intermediate points may however be easily found, which 
if accurately located and regularly distributed^ will render it \e\y 
easy to trace the required ellipse by hand. For this purpose, we 
therefore assume o' and n\ equidistant from C Each of these is 
the vertical projection of two points of the base. Revolve this base 
about A'B', till it becomes parallel to the vertical plane, and o' and 
n' will appear at o" and n". o'o" or n'n" — perpendicular to A'B' — 
will then be the true distance of the points at o' and n' before and 
behind the diameter A'B'. Hence from o' and n' draw projecting 



PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES. 55 

lines perpendicular to GL, making Os = Ot, and make sp = S7i =- 
tr = to = o'o" = n'n". Then o, n, r, and p will be four regularly 
distributed points through which, and the four previously found, 
the elliptical horizontal projection of the cone's base can easily be 
sketched by hand. 




£_ 




^E- 



E 



Fig. 30. 

To complete the horizontal projection of the cone, merely draw 
the tangents from V to the ellipse just drawn. The arc p^n be- 
tween these tangents, is invisible, being on the under side of the cone. 

Having now the complete projections of the cone, and the point of 
sight EE', the perspective can be found as in the previous problems. 

Example 10. — To find the Perspective of a Cone, Tvhose 
axis is oblique to both planes of projection. 

For further variety, the perspective plane is here taken (Fig. 31) 
as a third plane, PLP', perpendicular to the two principal planes of 



56 



LINEAR PERSPECTIVE. 



projection. It is, however, recommended to the student to solve 
this problem with two planes only, besides the auxiliary plane 
whose ground line is gl ^ also to solve the two preceding problems 
on three planes. 




L 



P* 



P 



Fig. 31. 

Let GL be the principal ground line, and VA and Y'A' the pro- 
jections of the axis of the cone. This axis being oblique to both 
planes of projection, the base which is perpendicular to it, is oblique 
to them both, also, and hence Avill appear as an ellipse, in both of 
the required projections, so that neither of these projections will 
show its real size. 

In all the simpler preceding figures, we have seen that at least 
one of the two p^'ojections shows two of the three dimensions of a 
solid in their real size ; so here, where neither of the required 
projections possesses this property, we must begin, as always^ with a 
projection upon an auxiliary plane so situated as to show upon it 
the real size of two of the dimensions of the cone. 



PROJECTIONS A.ND PERSPECTIVES OF CIRCULAR BODIES. 57 

Accordingly gl^ parallel to AV, is taken as the ground line of an 
auxiliary vertical plane, parallel to the cone's axis ; for two dimen- 
sions will appear in full size on such a plane. 

Now any number of different vertical projections of one fixed 
point, will be at equal heights above the ground lines of their 
respective vertical planes (44), and the two projections of the same 
point are always in the same perpendicular to the ground line (61) 
Hence make sV" = s'Y' and in the line W" perpendicular to gl 
Then make A A'' perpendicular to gl, and mA" = m'A', and V'A 
wiL be the real size of the cone's axis, projected on the auxiliary 
plane parallel to it. 

Next make c"k" perpendicular to V'A" and make A'^c" = A!'k" ; 
and draw Y"g" and \"k\ and Y"c"k" will be an auxiliary projec- 
tion of the cone, showing the true size of its altitude Y"A" and 
diameter G"k", 

From this auxiliary projection, make the horizontal projection as 
in the last problem ; Aa == Af== A"c" ; also br = re = td = th 
equal to h"h"\ and projected down from e" and h". 

Having thus completed two projections, the base in the required 
vertical projection is found as Y" was. Thus d'p' == h'n' == h"n ; 
a'o =f'g^ = A!' in, &c. Then sketching the ellipse a'h'e'c', and 
drawing the tangents to it from V, the required projections of the 
cone will be complete. 

The student can now proceed, as in previous problems where 
three principal planes at right angles to each other are used, to 
find the perspective of this cone AV — A'V on a perspective plane, 
as PLP', at right angles to both of the principal planes of projec- 
tion. 

Remark. — ^The operations of projection applied to the bases of 
the cones in the several preceding problems, are the same that 
would be necessary in finding the perspectives of isolated circles 
in similar positions. Hence separate problems upon circles hav« 
not been given. 

87. Here leaving Cylinders and Cones, whose surfaces are called 
single curved surfaces, because straight lines can be drawn in cer- 
tain directions upon them, we pass on to bodies having surfaces 
called double curved, such as a Sphere, on which no straight line 
can be draw^n.* 

* As most double curved surfaces in the arts are found on small objects, urns 
vases, &c., which may be drawn by the eye after their larger supporting or surround 



58 LINEAR PERSPECTIVE. 

Example 11.— To find the Perspective of a Sphere. 

88. This example is first taken, since a sphere is the simplest poa 
sible double curved surface, inasmuch as the section of it made by 
any plane, is a circle. 

From the fact just stated, it might at first be supposed that the 
perspective of a sphere would always he a circle ; but not so, though 
it would always appear (39) as a circle, from the given point of 
eight. For the visual rays from the apparent contour of a sphere 
will always form a cone, with a circular base which is this same con- 
tour. But the intersection of this cone with the perspective plane, 
which will be the perspective of the sphere, will not be a circle 
unless the perspective plane is perpendicular to the axis of this cone, 
that is to the visual ray from the centre of the sphere. This 
statement touches on the subject of conic sections, but the student 
can easily satisfy himself of its truth by placing a paper cone 
around a ball, observing its circle of contact with the sphere, sup- 
posing its vertex to be the place of the eye, and then intersecting 
the cone between its vertex and the sphere by planes in various 
positions. 

There are two quite different methods of determining the appa- 
rent contour, whose perspective constitutes the perspective of the 
sphere. The determination of this apparent contour forms the chief 
portion of the solution of the problem, and to it we therefore first 
particularly attend. 

For further and instructive variety, we will represent one of these 
methods of finding the apparent contour on two planes ; and the 
other on three planes. 

First Method^ — The projections of a sphere will evidently be 
two equal circles, whose centres will be in the same perpendicular 
to the ground line. Then, in Fig. 32, let the circle with O for its 
centre be the horizontal projection of a sphere, whose centre is at a 
distance behind the perspective plane equal to the distance of O 
from the ground line LL. The equal circle whose centre is O', i 
the vertijcal projection of the same sphere, and the height of O 
above LL is the height of the centre of the sphere above the hori- 
zontal plane. E and E' are the projections of the point of sight, 
and the vertical plane is taken also as the perspective plane. 



ing objects shall have been found, the two following problems may be omitted at the 
discretion of the teacher. 

* This method being chiefly valuable as an intellectual exercise in the conception 
of positions and motions in space, it may be omitted at the discretion of the teacher. 



PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES. 



59 



It is now evident that, if a vertical visual plane E05 be drawn 
through the centre of the sphere, it will cut a vertical great circle 




ye' 



1/ 



Fig. 32. 

from the sphere, to which two tangent visual rays may be drawn 
The points of tangency of these visual rays will, as such, be points 
of the apparent contour of the sphere. But to show these rays, the 
plane E05 must be revolved to a position parallel to one of the 
planes of projection. Let it be revolved about the horizontal 
diameter, ab^ of the sphere, till it becomes parallel to the horizontal 
plane of projection. The vertical great circle will evidently then 
appear in the great circle Ac"^, on ah as a diameter. The eye is at 
the vertical distance E'n below the level, O'/i, of the centre of the 
sphere ; hence, if the highest point of the vertical circle revolve to 
the right, as shown by the arrow, EE' will be found after revolu 



60 LINEAR PERSPECTIVE. 

tion at e"^ at the left of the axis of revolution 5aE, and at a pei 
pendicular distance from it equal to E'/i. 

This done, e'V and (^'d]' are the revolved positions of the desired 
tangent visual rays, and their points of tangency, q" and d\ with 
the circle d"c!'h^ the revolved positions of two points of the appa- 
rent contour. By revolving the plane containing d' and dl' back to 
its vertical position Ea5, d' and dl' will revolve back about ah as an 
axis in arcs whose horizontal projections are d'c and d"d^ perpendi 
cular to ab^ and will give e and dj as the horizontal projections of 
these two points of the apparent contour. 

To find the vertical projection of c, for example. Consider, first, 
that it must be in a perpendicular to LL, through c, and that it is 
at a height equal to cd' above the level O'n of the centre of the 
sphere. Hence on c-d make dk = c"c, and d will be the vertical 
projection of the point of apparent contour whose horizontal pro- 
jection is c. The vertical projection of c?, not shown, in order to 
simplify the diagram, will be below O'/i at a distance equal to d"d. 

To find any other points of apparent contour. Intersect the 
sphere by any other vertical visual plane as E^', which cuts from it 
the small circle whose revolved position, about pq as an axis, is 
f'g"q. In this revolution, EE' will appear at d" ; Ee'" being equal 
to Wn and perpendicular to Eg'. The revolved visual rays d' f" and 
d"g" contained in this plane, give the points of contour f" and g\ 
whose true positions, found as before, are /' and g. The vertical 
projection of g,g'\ found also as before, is g\ Any other points of 
contour may be similarly found. 

Finally hh\ the point of contact of a tangent vertical visual plane, 
is a point of apparent contour. A similar point may be likewise 
found near d. Also two tangents from E', as E'm', to the vertical 
projection of the sphere will be the vertical traces of visual planes 
perpendicular to the vertical plane, as EA is to the horizontal plane. 
Hence their points of contact, as m'm, will be real points of apparen* 
•ontour on the vertical great circle through 00' and parallel to th 
vertical plane of projection. 

[This comparatively tedious construction, which shows how 
greatly geometrical problems increase in complexity as we leave 
simple plane sided solids, show^s therefore why elementary works 
on perspective so often confine themselves wholly to such solids, and 
to plane figures.] 

After finding the points of apparent contour, the construction of 
their perspectives is the work ot* a moment. Thus, to find the per- 
spective of cd. Ec -E'c' is the visual ray from this point, and by 



PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES 61 

(58) this ray pierces the perspective plane at C, which is therefore 
the perspective of cc' . By finding the perspectives of the other 
jioints of apparent contour similarly, and joining them, Ave shall 
Lave the perspective of the sphere as seen from EE'. To remove 
the persj^ective from the projections ; as before, translate the per- 
spective plane forward to L'L', and c' an equal distance, and pro 
ceed as in (Ex. 3) to find the perspective of cc' at C instead of at C 

Remarks. — a. The method just explained is, from its nature; 
called the method by secant visual planes. 

J). The student, as soon as he clearly conceives of the positions 
and motions explained in the preceding solution, and represented in 
Fig. 32, will be able to see that the vertical visual planes, as Ea5, 
might have been revolved in two other ways. First: around a 
vertical line at E till they should be parallel to the vertical plane 
of projection. Then the vertical circle ab would appear vertically 
projected in a circle, to which tangent visual rays could be drawn 
from E'. Second: These planes might have been revolved to a 
similar position, each, about the vertical diameter, as at O, of the 
circle contained in it. In this case EE' would revolve to a new 
position. 

c. Again : instead of vertical visual planes, visual planes migh( 
have been passed perpendicular to the vertical plane of projection. 
Any line through E' and the vertical projection of the sphere would 
be the vertical trace of such a plane, and it might be revolved in 
three ways, analogous to the three ways of revolving Ea5, in order 
to show the circle and tangent visual rays contained in it. 

The completion of the constructions thus suggested, is left as a- 
valuable exercise for the student. 

Second Method. — In this method we make use of these principles. 
1° : Tliat when a plane is tangent to the surface of a cone, it is tan- 
gent all along a straight line or element of that surface, from its 
vertex to its base. 2" : That such a plane therefore contains the 
vertex. 3°: That, therefore, if such a plane also contains a point 
m space, it will contain the straight line joining that point with the 
cone's vertex. 4° : That the line cut from the tangent plane by the 
plane of the cone's base is tangent to that base at the foot of the 
element of tangency (85,3°). 5° : That a cone may be circum- 
scribed tangentially around a sphere and Avill then have a circle of 
contact with the sphere. 6°, and lastly : That the point where the 
dement of tangency oi 2^ plane and cone intersects the circle of tan- 
gency of the same cone vnth a sphere., will be the point of contact 
of that plane with the sphere. 



62 



IJNEAR PERSPECTIVE. 



If now the given point in space (3^) be the point of sight, the tan 
gent plane will also be a visual plane, and its point of contact will 
be a point of the apparent contour of the sphere, whose perspective 
will be the perspective of the sphere. 




Now in Fig. 33, let Ym-A'B'd' be the given sphere, PQP' the 
perspective plane, and EE' the point of sight. Assume the circle 
A'W—Au^t as the circle of contact of an auxiliary tangent cone 
(5<^). The tangents V'A' and VB' complete the vertical projection 
of this cone. EV-E'V, the visual ray through Y V, the cone's 
vertex, pierces A'R', the plane of its base, at R'R-R' being found 
first and then projected horizontally at R (3°.) Hence by (4°) R^ 
and Ri^ are the traces of the tangent planes, to the opposite sides 
of the cone, upon the plane of the cone's base. Then by (4°) and 
(6°) tt' and uu' are points of contact of two visual planes with the 
sphere, and are therefore points of the apparent contour of the 
sphere. 

This being determined, we find the perspectives of tt' and uu as 



PROJECTIONS AND PERSPECTriVES OF CIRCULAR BODIES. 63 

in previous cases where three planes have been used. Thus, the 
visual ray ^E-^'E' pierces the perspective plane at hb' ; which is 
translated parallel to GL, with the perspective plane, to the new 
position PiP/ then revolved about QiPi' as an axis, into the verti 
cal plane at T, which is the perspective of tt' . The perspectives of 
)ther points of the apparetit contour may be similarly found. 

As the eye is placed, in this problem, in the horizontal plane 
'E', through the centre of the sphere, the apparent contour of th 
sphere is evidently a vertical circle, hence nin^ a straight line 
through t and w, and which will be perpendicular to YE, is its hori- 
zontal projection, n and m, being on the horizontal great circle ot 
the sphere, are vertically j^rojected at n and rn! . Two points are 
horizontally projected at c, at t and at u. Those at c are on the 
vertical great circle A'B'c?', and are vertically projected at c' and d' . 
r' and s' are in the lines t-t' and u-u' and as far below the line 
m'?i', as t' and u' are above them. 

Remarks. — a. This second method is, from its nature, called the 
method of tangent visual planes. 

1). In order to familiarize the learner more effectually with this 
beautiful method of tangent visual planes, located by the use of 
auxiliary tangent cones, we will now apply it to an object of 
another and very different form, and with the use of three planes 
of projection. 

Among the comparatively few large double curved surfaces occur- 
ring in the mechanic arts, whose perspectives need to be accurately 
constructed, are Domes, and concave Spires, &c., whose perspectives 
may be found as follows. 

Example 12. — To find the Perspective of a concave 
Cupola-Roof. 

Let the figures with centre A and vertex A', Fig. 34, be the projec- 
tions of the cupola roof; PQP' the original, or real position of the 
perspective plane, and EE' the point of sight. The construction of A'' 
is, after previous similar constructions, sufficiently indicated in the 
figure. Then assume B^T-B'C as the circle of contact of an auxiliary 
cone, tangent to the inside of the cupola. Drawing Q'v' tangent 
to the cupola at C, we find v* the vertex of this cone. Then AE- 
w'E', the visual ray from this vertex, pierces the plane of the cone's 
base at R'R (Ex. 11), hence R^ and RT are the traces, on this 
plane, of two planes which are tangent to this cone on elements A^ 
and AT (not drawn) and hence to the cupola at the points tt' and 
TT' (Ex.11 ; 6°). The perspectives of these points are t" and T'', 
found by making c'T' =Qc, &c. (Ex. 6, Rem. h.) 



64 



LINEAR PERSPECTIVE. 



At the base of the cupola is a round edged band, three points of 
which, mm! ; nn' and / it is sufficient to find in perspective, as aV 




m% n"^ and r" , Through these points the perspective of the cupola 
can be sketched. 

Examples for Practice. 

1. In Example 9, let the axis of the cone be parallel to the horizontal 
plane only, and find its perspective. 

2. Find the perspective, either by two, or by three planes, of a regular 
octagonal prism, each side of whose upper base is ttie diameter of a vertical 
semicircle. 

3. Make the perspective of a simple octagonal, or circular, summer-house. 
(The projections, made in pencil on a temporary sheet pinned to the board, 
can be on such a scale that the persjiective will occupy a plate of the size of 
those in this volume, ) Minor details can be sketched by hand after deter- 
mining points have been found in perspective. 

4. Make the perspective- (as above) of a house porch and steps. 



i'ERSPKCTIVBS OF SHADOWS Gfi 



CHAPTER VII. 

PERSPECTIYES OF SHADOWS. 

» General Principles and Illustrations. 

89. Perspectives of shadows, like those of objects, are readily 
found from their projections, by the method of visual rays already 
explained. 

But shadows, being obviously not independent of the bodies cast- 
ing them, require a little separate preliminary study, to show how 
they are found when those bodies are given. 

90. The shadow of a body on any surface, is that portion of that 
surface from which light is excluded by the body. 

A shadow is known when its bounding edge, called the line of 
shadow^ is known. 

91. Rays of light from a very distant source, as the sun, fall 
upon any terrestrial object m parallel straight lines. 

92. Any ray which is intercepted by the given body will, evi- 
dently, if produced through the body, pierce the shadow within its 
boundary. Any ray, not intercepted by the body, will evidently 
pierce the supposed surface containing the shadow, beyofld the edge 
of the shadow. Hence the line of shadow is the shadow of that line 
on the given hody^ at all points of which the rays are tangent to the 
body. 

This line of contact of rays, separates the illuminated from the 
unilluminated portion of the given body, and is called the line of 
shade. 

93. Since the line of shadow is thus the shadow of the line of 
shade, the latter must always be found first. 

The line of shade, from Avhich shadows are determined, is found 
in the same general manner as the line of apparent contour, from 
which perspectives are determined ; viz. by inspection on most 
plane sided bodies, and by the aid of tangent rays of lights or tan- 
^ent planes of rays oflight^ on curvtd surfaces. 

5 



66 



LINEAR PERSPECTIVE. 



94. Practically, shadows are found, a point at a time, and any one 
point in a line of shadow is where a ray of light, from some point in 
the line of shade of a given body, pierces the surface receiving the 
shadow. 

Hence it is obvious that the form of a shadow will depend both 
on the form of the body casting it and that of the surface receiving 
it', and also on the direction of the light ; while the method of find 
ing it will depend only on the form of the surface receiving it. 

95. Finally; To find a shadow^ we must have given, by their 
projections, \st. The body casting it ; 2nd, The surface receiving it ; 
^rd. The direction of the light. These given, we may then con- 
struct, \st. The line of shade on the given body ; 2nd. The shadow 
determined by that line of shade. This done, we can at last con- 
struct the perspective of the shadow. 

Problems of perspectives of shadows being thus obviously some- 
what tedious and complex, only a few simple and generally useful 
ones are here inserted, as an introduction to the subject. 

The shadows which most frequently occur in perspective drawings 
such as are made largely for pictorial effect, are the shadows cast 
by lines in various positions, on the ground, and on the walls and 
roofs of buildings. The following principles and examples, therefore, 
give elementary illustrations of the operations necessary in finding 
such shadows. 

96. Let AB, Fig. 35, be a slender vertical rod or wire, and let 
LR represent a ray of light drawn through its upper extremity, B, 

and piercing the horizontal 
plane GH at R. Then R 
will be the shadow of the 
point B. But the point, as 
A, ill which a line meets a 
surface, is a point of the sha- 
dow of that line on that sur- 
face. Hence AR is the sha- 
dow of AB on the horizontal 
plane GH. But BA being 
vertical, AR is also the hori- 
zontal projection of the ray 
BR. Hence, the shadow of 

a vertical line on the horizontal plane is in the direction of the 

horizontal projection of the light. 

97. By operating in a precisely similar manner upon a line per- 
pendicular to the vertical plane GV, it will be found that the sha- 




G<; 



Fig. 35. 



PERSPECTIVES OF SHADOWS. 



67 



dow of such a line upon the vertical plane, will be in the direction 
of the vertical projection of the rays of light. 

98. Since the rays of light are parallel, it is clear that the shadow 
of a vertical line on the vertical plane^ will^ itself he a vertical 
line ; likewise, the shadow of a horizontal line on a horizontal plane, 
will be parallel to the line, and, generally, for the same reason, if a 
line be parallel to any plane, its shadow on that plane will be 
parallel to the line itself. Hence, also, the shadows of parallel lines 
on the same plane^ will he parallel to each other. 

99. If the same line casts a shadow on both planes of projection, 
the shadows on the two planes must meet the ground line at the 
same point. Thus in Fig. 36, let AC be a vertical line, long enough 
to cast its shadow partly on each plane of projection. Then R, 
where the shadow All leaves the horizontal })lane GH, must be the 

beginning of the shadow^ RT on 
the vertical plane. For all the 
rays CT, BR, &c., from points on 
AC, being parallel, form a plane 
called a plane of rays^ of which 
AR and RT are the traces, and it 
has already been shown (YS), that 
the two traces of the same plane 
must meet the ground line at the 
same point. 

Finally, in general, if the sha- 
dow of any line, as the top edge of 
a roof, falls on both of any two intersecting surfaces, as the front 
and side of another building, these two shadows will meet at a 
common point on the edge dividing those surfaces. Hence, %ohen 
we have the complete shadow on one such surface^ this common 
'point gives us one point of the shadow on the other surface. 

Example 13. — To find the Perspective of the Shado^w of 
a Square Abacus upon a Square Pillar. 

The method of two planes is here employed, GL, Fig. 37, being 
the first, and G'L' the second position of the ground line. The 
construction of the perspective of the pillar and its cap (abacus) is 
not shown, it being exactly like many previous constructions. 
Also, no more of the vertical projection of the object is made than 
is necessary in finding its perspective, and shadows. 

Rays of light, like other lines, being indicated, in position, by 
their projections, let A^ and A'L' be the projections of the ray 
through the front right hand upper corner, AA', of the aba«3us 




h:^ 



Fig. 36. 



68 



LI^STEAE PEKSPECTIVE. 



This ray pierces the horizontal plane at l^ whoso vertical projection 
which must be in the ground line (45), is L'. Therefore the point 




AE 



Fig. 37. 



'tself is at l^ which is therefore the shadow of AA' on the horizontal 
lane. 
The shadow of the lower front edge, ak.-a'h\ of the abacus, 
upon the front of the pillar, will be parallel to itself (98). When 
the direction of a line is known, one point in it is sufficient to 
determine it ; hence, to find this shadow, it is only necessary to 
pass a ray, as cb-c'd\ through any point, as cc', of the lower front 
edge of the abacus,.and to find wliere this ray pierces the front face 
of the pillar, as at h^d' . In this case, by drawing the ray through 
5, in plan, the point of shadow, 5,c?', is made to fall on the right 
hand vertical edge, h-b'f\ of the pillar. A line through d\ and 



J 



PERSPECTIVES OF SHADOWS. 69 

parallel to a'h\ will be the vertical projection of the shadow of ajA- 
a'h' . Drawing the visual ray c?'E', its intersection, D, with DF, the 
perspective oih' f\ will be the perspective of c?'/ and as the shadow 
is parallel to the perspective plane, its perspective will be parallel 
to itself (69, a), that is a horizontal line through D. 

Next, drawing the ray ah-a'h\ we find hh\ the shadow of aa' 
the lower, front, left hand corner of the abacus, on the side surface, 
gn^ of the pillar, whose vertical projection is a line through h' equal 
and parallel to h'f. The visual ray, AE-A'E', from hh' gives its 
perspective, e. Then eo is the shadow of a small portion of aA- 
a'h' upon the left side of the pillar. 

Now for the shadow on the horizontal plane. The shadow of the 
point AA' is ?, where the ray A/-A'L' pierces that plane. The ver- 
tical projection of ^ is L' (45), and the visual ray, ZE-L'E', therefore 
gives M as the perspective of the point l\J , In the same way, find 
the shadows of the points Kf)' and 55', and join the latter shadow 
with F. The shadow of A'-AK will be parallel to that line, and 
will begin at I. Hence, as will fully appear on making the construc- 
tion, L'E' will also be the vertical projection of the visual ray from 
the shadow of K. Hence ME', up to DF, is the visible portion of 
the perspective of the shadow of A'-AK. 

Example 14. — To find the Perspective of the Shado"W of 
any triangular Pyramid upon the Horizontal Plane. 

In this problem we shall employ the simple principles, that the 
shadow of the point where any number of lines meet is the point 
where the shadows of those lines meet ; and that the point in which 
a line pierces the horizontal plane is a point of its shadow on that 
plane. 

The method by two planes is employed, and the construction of 
the perspective of the pyramid, being the same as in many previous 
problems, is briefly indicated in the diagram, only, Fig. 38. 

Let ABC be the plan of the base of the pyramid, and Y, that of 
its vertex. V-A'B'C is the vertical projection of the pyramid 
This vertical projection, being shown in full on the original positioi 
of the vertical, or perspective plane, only its points, A"B"C" and V", 
are shown, in the same relative position, on the translated position 
of the same plane, whose ground line is G'L'. In fact, after becom- 
ing quite at home in the subject of perspective, the student will see 
that A'B'C'-V might have been omitted altogether; and, in gene- 
ral, that often only points, and not lines, of the projections of 
objects. need be shown, in order to find their perspectives. 

Having, as in previous problems, found v-ahc, the perspective of 



70 



LINEAR PERSPECTIVE. 



the pyramid, draw the ray of light VK-V'R' which pierces tht 
horizontal plane at R, projected back from R' in the ground line. 




Fig. 38. 
Than R is the shadow of W on the horizontal plane, and A and 



PERSPECTIVES OF SHADOAVS. Yl 

C being their own shadows — (96 — and the second principle above 
stated — )RA and RC are the shadows of VA and VC. Then, 
drawing the visual ray RE-R^E', we find r for the perspective of 
RR" ; R" being the vertical projection, R', in its second position. 
Hence ra and re are the perspectives of the shadows RA and RC, 
which limit the shadow whose perspective was required. 

Example 15. — To find the Perspective of the ShadoTV of 

Dormer WindoTW upon a Roof. 

In this concluding example of shadows, found by primitive 
methods, we will, for further variety, employ the method by three 
planes. Again, this example involves the shadows of lines in three 
different positions, upon a slanting surface, and affords the most 
instructive variety with the fewest lines. Moreover, as the sha- 
dows of lines are determined by the shadows of points in them, and 
as the shadow of a point is the same — and similarly found — whether 
the point be on a straight line or curve, a careful study of this and 
the two preceding examples should enable the student to find the 
projections and perspective of any ordinary shadow. 

First, now, in Fig. 39, to find th^ projections of the windows and 
roof. To avoid unnecessary lines, only a small portion of one slope 
of a roof is shown, of which ABCD is the plan, CDC'D'" is the 
auxiliary elevation, showing the true size of the front of the dor- 
mer, and A'B'C'D -C"D", found as in Ex. 10, is the principal 
elevation. G and F are the plans of the vertical edges of the 
dormer, whose true heights I^G" and J"F" appear in vertical pro- 
jection at I'G' and J'F'. The vertical projection, E', of the peak 
of the gable, FEG, is found on the projecting Hne ENE', by laying 
off NE' equal to its height, N"E", above the horizontal plane. 
Then draw E'G' and E'F'. NK' parallel to B'C", is evidently 
the trace on the roof, of a vertical plane through the ridge 
EK-E'K', which therefore meets the roof in this trace at K', 
whose horizontal projection is K. The points HH' and LL' are 
imilarly found, and then joined with KK'. 

Next, to find the projections of the shadows on the roof. Le 
FP and F'P' be the projections of a ray of light to which all the 
other rays are parallel. The shadow of the vertical edge, F-F'J', 
will fall in FS-J'S', the trace on the roof of a vertical plane of rays 
(99) through that edge. 

The ray FP-F'P' meets this trace at P', which is then horizon- 
tally projected at P, and FP-J' P' is the shadow of F-J'F'. Then 
PP' being the shadow of FF' (94), and LL' being its own shadow 
(96), LP-L'P' is (2:eometrically, for this shadow is unreal), the sha- 



72 LINEAR PERSPECTIVE. 

dow of FL-F'L'. The shadows of parallel lines, on the same plane. 




Fig. 39. 



PERSPECTIVES OF SHADOWS. 73 

bemg parallel (98), KR-K'R' the shadow of EK-E'K', is parallel to 
LP-L'P' and is limited at R by the ray ER. R' is then projected 
from R. Finally, by drawing RP-RT', we have the shadow of 
EF-ET'. 

Lastly, to find the perspective of the roof and shadows, whose 
projections have just been completed. Let h"A's be the original, 
and a'd' the translated position of the perspective plane ; and let 
00' be the point of sight. This construction scarcely needs any 
explanation, exactly similar ones having been often fully explained 
already. One or two points only are mentioned, to acquaint the 
learner with the abbreviations which are made in the construction 
of the figure. To find e, for example. Draw the ray EO-E'O' and 
from its intersection, n^ with the perspective plane, draw nn\ parallel 
to the ground line ; then make n'e^=zn"A\ and e will be the perspective 
of EE', since this is obviously equivalent to translating n" to n"\ 
and revolving it, as in the previous imabridged constructions. 

Having found c?, the perspective of DD", in the same way, also^ 
and g^ the lines dd\fj and gi can immediately be drawn perpendi- 
cular to the ground line, since they are the perspectives of vertical 
lines, dd" is limited at d\ simply by drawing D'O' to w, and ud"^ 
parallel to the ground line, fj and gi are limited by their intersec- 
tion with ab. a is the perspective of the point AA', which is its 
own perspective, it being in the perspective plane, and a'a^A'A. 

The perspectives of the points of shadow are found in the same 
manner. Thus, to find ^, draw the ray PO-P'O' to qq\ and the 
line of translation q' p\ and mskQ p' p—A*q^ which will give ji?, the 
perspective of PP'. Drawing jp^ finding r, the perspective of 
RR', as p was just found, then drawing jor, and rJc^ we sh?il have 
the complete perspective of the shadow on the roof. 

Example 16. — To find the Perspective of a convex, 
four-sided Dome, Flag-staff, and their Shadows. 

ABCD-F-A'B'F' are the projections of the dome ; and A-A'S', 
of the staff — the elevation on the translated position^ on G'L', 
of the vertical and perspective plane. E (indicated by the 
intersection of M^ and A^) and E' (indicated by the intersec- 
tion of Wm and F'/), the projections of the point of sight/ are 
both beyond the limits of the page. 

FO-F'O', assumed as the ray of light from the apex, FF', of 
the dome, gives (Ex. 14) as the shadow of FF' on the horizon- 
tal plane. Then FE-F'E' and OE-O'E', being the visual rays 
from FF' and from 00', we find, as before,/ for the perspec- 
tive of FF', and o for the perspective of the point of shadow. 



74 



LINEAR PERSPECTIVE. 



0. Operating similarly for other points, we find the perspec- 
tive, abcf, of the dome ; llco, the perspective of the shadow of the 
edge BHE-B'H'F' on the horizontal plane ; and hm, that of the 
like shadow of the staff A-A'S'. 

To find the shadow A:N'H-A'N'H' of the staff on the front of 
the dome : Its horizontal projection, AH, is parallel to FO. H 
is projected at H' on F'B'. Other points, as W^', are found 




B^ig. 39a 

by taking planes, as NP, parallel to the vertical plane, and cut- 
ting the dome in a horizontal line, NP-NT' (P' projected from 
P),^on which N is projected at W. The perspective n, of ISTN', 
is then found, as before, as the point r, n, where the visual ray 
NE-N'E' pierces the perspective plane. 

Examples for Practice'. 

1. Construct Example 16 by the method of three planes, and on a large 
scale. (See carefully Examples 4 and 15.) 
3. In Example 16, let the dome be a hemisphere resting on a square prism. 



^ 



GENERAL PRINCIPLES AND ILLUSTRATIONS. ^5 



PART II. 
DERIYATIYE METHODS 



CHAPTER I. 

GENERAL PRINCIPLES AND ILLUSTRATIONS. 

100. In all the problems of Part I., we have found the perspec- 
tive of every point by one and the same primitive and naturai 
method^ which consists in finding where a visual ray (actually 
represented) through any given point, pierces the perspective plane. 

This method is primitive, and peculiarly the natural one, because 
it manifestly embodies the simplest geometrical definition of the 
perspective of a given point, viz. that it is where a visual ray from 
that point pierces the perspective plane (35). 

It is true, that in the practical application of this method, having 
revolved the perspective plane directly back into the horizontal 
plane, a difficulty arose, as in Figs. 16 and 17, from the confound- 
ing together of the projections and the perspective in one place on 
the paper. This difficulty led, ^rs^; to the translation forward of 
the perspective plane, till it could be revolved back into the hori- 
zontal plane so as to bring the perspective below the projections, as 
in Fig. 19; second: to the use of three distinct planes, as in Figs. 
22, etc., where the difficulty of confusion of figures was obviated 
Bost completely. But these merely particular graphical methods 
f applying the method of visual rays, evidently do not alter the 
method itself, and we repeat, that all the problems of Part I. were 
solved by the primitive method of finding where visual rays, actu- 
ally represented, through given points, pierced the perspective 
plane. 

101. AU problems whatever, in perspective, might be readily 
solved in this simple and beautiful manner ; but by inspection of 
the perspectives thus found, certain peculiarities may be discovered, 
which, on examination, lead to other methods, hence called deriva- 



76 LINEAR PERSPECTIVE. 

tive ; or, because the visual rays are no longer represented, com 
paratively, artificial methods. 

The advantages of knowing several methods, which will soon 
appear, are chiefly two : 1" Abbreviation of the operations of con- 
struction. 2* Provision of checks upon inaccuracy. 

It has already been showm by experimental proof, in previous 
constructions, that any lines, whether vertical or horizontal, which 
are parallel to the perspective plane, have their perspectives parallel 
to each other, and to the lines themselves. Also that the perspec- 
tives of all lines which are perpendicular to the perspective plane, 
meet at the vertical projection of the eye. (Fig. 16.) These two 
results have also already been separately proved to be true, (Ex. 
l.i?ems.a,5.) but by now considering them in connexion with a few 
others, we shall arrive at a body of principles by which perspec- 
tives of objects can be found by the derivative methods, which it 
is the object of this second " part " to explain. 

102. In standing on a vast plane, such as a natural plain, its 
remotest visible limit appears as a horizontal line on a level with 
the eye. The reason of this is evident from Fig. 40. Let E be the 



3.. 



E 



Fig. 40. 

place of the observer's eye, looking forward in the direction ER, 
parallel to the ground HP. In taking successive points on the 
ground, as a, 5, and P, at greater and greater distances from the ob- 
server, standing at H, the visual rays oE .... PE, &c., become more 
and more nearly horizontal, and finally, when a ray comes from an 
indefinitely remote- point on the ground, its direction cannot be dis- 
tinguished from that of the horizontal visual -ray ER. Hence, as the 
apparent position of objects depends on the direction of the visual 
rays entering the eye from them, the very remote limit of any level 
plane appears as a horizontal line, on a level with the eye. 

103. The indefinitely remote limit of a natural plain, or horizon- 
tal geometrical plane, is called the horizon ; hence a line parallel 
to the ground line, and through the vertical projection of the point 
of sight, is the perspective of the horizon. 



GENERAL PRrSTCIPLES AND ILLUSTRATIONS. 77 

Such a line is called the horizontal line^ or the horizon of the pic- 
ture. Like reasoning applies to any horizontal plane. 

104. It follows from this, that the remotest visible limit of aU 
lines in such a plane^ that is of all horizontal lines, will appear to 
be in the " horizontal " line, which represents the remotest limit of 
this plane. 

105. Now the remotest visible limit of the plane supposed, is 
literally its vanishing line^ and, likewise, the remotest visible point 
of any line in that plane, is its vanishing point. The representation 
of this line, or point, on the perspective plane, is the perspective of 
such line or point, and is, according to the last two articles, a line 
or point on the perspective plane, and at the height of the eye. 

106. The perspective of a vanishing line or point being of con- 
stant use in the construction of perspectives, while the original 
indefinitely distant real vanishing line, or point, is not, the former 
is, for brevity, itself termed the vanishing line, or point. 

Hence we have these principles : 1°. The vanishing line of any 
horizontal plane, is a horizontal line, drawn on the perspective plane 
and at the height of the eye, 2°. Any horizontal line has its 
vanishing point in the '•''horizontal'''' line. 

107. Similar reasoning might be applied, and with corresponding 
conclusions, to vertical, or oblique planes, but as we do not find in 
Nature real planes of indefinite extent, everyway, in these positions, 
it will be sufiicient to consider the vanishing points of ^mes, only, in 
any direction. 

108. The visual ray from the indefinitely distant, or remotest 
visible limit of an unlimited line, will evidently appear to be parallel 
to that line, and the intersection of this ray with the perspective 
plane, is the perspective of that remote or real vanishing point. 
This intersection itself (106) is practically called the vanishing 
foint, in making perspective drawings ; and will be so called in the 
following pages. Also, a visual ray which is parallel to one line, is 

arallel to all others, which are parallel to that one. Hence to find 
the vanishing point of any line or group of parallel lines, we have 
the following rule. Find where a visual ray, parallel to the given 
lines, pierces the perspective plane ; the point thus found will he the 
required vanishing point. 

109. Illustration. Let PP, Fig. 41, represent the perspective plane, 
and L,L,L, three parallel lines in any direction. These lines will appa- 
rently meet, and so maybe considered as meeting, at an indefinitely 
great distance, and the visual ray VE from their distant apparent 
intersection, will, for any short distance, as EV, be sensibly parallel 



78 



LINEAR PEESPECTIVE. 



to them. But V, the intersection of this ray with the perspective 
plane, is the perspective of that intersection. That is, V is the 
vanishing point of L,L,L, 





P 




~< — 

T 


^ 




/ 


/ 


\L^^^\ 




/, 


1 


\N^ 


/ 


7/ 


1 


^\^^^\^ 


'a 

1 


// 


pj 


^>s^^^ 


--./ 


/ 


y^ 






> 





^% ^ 



Fig. 41. 

110. By (100) if lines intersect at a point, its perspective will 
be the intersection of tlieir perspectives. Hence the perspec- 
tives of parallels will meet at their vanishing point. Again : 
The point where a line pierces the perspective plane is a point of 
its perspective ; for the visual ray from that point there pierces 
that plane. Thus, if L,L,L pierce the perspective plane at a, h, 
and c, dV, 5V, cY, are their perspectives and meet at V. 

Example 1. — ^Let it be required to find the vanishing 
point of several Telegraph Wires -which go over a hill. 

In Fig. 42 let AA' and BB' be two successive poles, carrying two 
wires. AB is the plan of both of these wires. Let CC and DD' 
be another pair of poles, of a line of single wire, and let EE' be the 
position of the eye. Then EV, parallel to AB or CD, and E'VS 
parallel to A'B' or CD', are the projections of the visual ray, parallel 
to these wires, and therefore giving the perspective of an 
indefinitely remote point upon them. This ray meets the perspec- 
tive plane at V (58), which is therefore the vanishing point at 
which the perspectives of the wires will meet. 

111. From the general case just considered, in illustration of the 
general principle of (108) let us proceed to find the location of the 
vanishing points of groups of parallels, having particular positions 
with respect to the perspective plane. 

First. It follows directly from the rule (108), that all lines which 
are parallel to the perspective plane have no vanishing point. 
Hence their perspectives will be parallel to themselves. That is, 
the perspectives of vertical lines, for example, will be vertical, as 



GENERAL PRINCIPLES AND ILLUSTRATIONS. 



19 



Been in (Fig. 16, etc., Part I.) Also, if lines are parallel to the 
ground line, their perspectives will be parallel to the ground line, 
as also seen in Fig. 16. 




Fig. 42. 



Second. It also follows from (108), that all horizontal lines nave 
their vanishing points in the horizontal line, or horizon (103). 

112. In particular, among horizontal lines, we notice those which 
are also perpendicular to the perspective plane ; and those which 
make an angle of 45° with the perspective plane. The former are 
called perpendiculars, and the latter, diagonals. 

Example 2. — To find the vanishing point of a Perpendicu- 
lar, and of a Diagonal. 

See Fig. 43, where Dc is the ground line, EE' the point of sight 
and D'A the horizontal line. 

By (48), when a line is perpendicular to the vertical plane, its 
vertical projection is a point, and its horizontal projection, a line, 
perpendicular to the ground line. Therefore ab is the horizontal, 
and a' the vertical projection of a perpendicular, at the height aa' 
above the horizontal plane. Likewise Ee is the horizontal, and E' 
the vertical projection of a visual ray, parallel to ab-a\ This 
visual ray pierces the perspective, or vertical plane at E', which 
is therefore the vanishing point of ah-a' and of all perpendiculars 
(108) while EE' remains as the place of the eye. 



80 



LINEAR PERSPECTIVE. 



113. The point E', the vertical projection of the point of sight, is 
usually known among artists as the centre of the picture , since in a 
picture of equal interest throughout, it should be in the centre, of 
the horizontal width, at least, of the canvas. Therefore we say that 
the vanishing point of perpendiculars is at the centre of the picture. 
E is often called the station point. 

114. To return now to the diagonal. By (51), when a line is 
parallel to the horizontal plane only, its 'vertical projection is paral- 
lel to the ground line, hence (112), making ac = ab^ abc = acb = 
45°, and he will be the horizontal projection of a diagonal through 
the point 6, a' and a'c' will be its vertical projection. Then ED- 
E'D' is the parallel visual ray which pierces the perspective plane 
(58), at D'. Hence D' is the vanishing point of hc-a'c' and of all 
other diagonals (108) parallel to Ic-a'c', 



d' -e' 


* ?- 

\ ' A 


! !£ a 


^ 


\ 

\i 


e 



Fig. 43. 

115. Observe in Fig. 43, that Ee = eD = E'D', that is, the dis- 
tance from the centre of the picture to the vanishing point of dior 
go7ials is equal to the distance of the eye from the perspective plane. 
Hence, having either E or D' given, with E', we can find the other of 
these points. Thus, having E' and D,' make eE = E'D' which gives 
E; and having E and E' given, make E'D' = Ee, which gives D\ 

Example 2a. — To find the Perspective of a Perpen- 
dicular and a Diagonal. 

116. Two points determine a straight line. But by (109) the 
vanishing point of a line, and by (110) its intersection with the 
perspective plane, are points of the perspective of that line. 
Hence, the perspective of a straight line is a straight line which 
joins its vanishing point with the point where it pierces the per- 
spective plane. 



\ 



GENERAL PRINCIPLE AND ILLUSTRATIONS. 



81 



Thus, in Fig. 44, the perpendicular db-a' pierces the perspective 
plane f.t a'/ and the diagonal, at c' y hence, if we draw a'E', it will 
be the perspective of this perpendicuhir, and if we draw c'D', it 
win ^'< 'he perspective of the diagonal, hc-a'c'. 



0' 


^ ! 


\ IP/ 


u \lr 


\ 1 
\ 1 
\ 
\ 

\ 

\ 1 

N 1 

\^ 1 


- 



Fig. 44. 



117. By (110) if two lines intersect at a pointy their perspec- 
tives will intersect at the perspectives of that point. Hence in 
Fig. 44, B, the intersection of the perspectives of the perpen- 
dicular ab-a' and diagonal hc-a'd , is the perspective of the point 
h, a from which both of these lines originated. 

Examples for Practice. 

1. In Fig. 42, find the perspectives of the three given lines. 

2. In Fig. 44, \st — let ac be laid off from a to the left, instead of, as now, 
to the right ; 2(?— let hc-aJd be to the left of EE'. 

Particular Derivative Methods. 

118. It is now apparent that, by the principles of (116) and (117) 
the perspective of any point, and hence of any object, can be found 
without the use of any visual rays. 

Derivative methods, then, consist in substituting for the visual 
ray from any given point, any two lines containing that point ; and 
in finding their perspectives, by joining their intersections with the 
perspective plane, with their vanishing points (116). The intersec- 
tion of the perspectives of these lines will then be the perspective of 
the given point (117). 

119. Since all parallel lines have the same vanishing point (108). 



82 LINEAR PERSPECTIVE. 

it will obviously abridge the constructions to use auxiliary lines in 
parallel sets. This being clear, it further appears, that no auxiliary 
lines are so universally simple and convenient as diagonals and per- 
pendiculars / first : because the centre of the picture, which is 
always given, being the vanishing point of perpendiculars, no 
vanishing point need be constructed for them ; second, because the 
distance D'E' from the centre of the picture to the vanishing point 
of diagonals is equal to the distance, eE, of the eye from the per- 
spective plane. Fig. 43 ; so that if the latter is given, the former is 
immediately known, and if it is not given, E'D' cati he assumed at 
pleasure. 

Foremost therefore amo7ig derivative methods^ is the method of 
diagonals and perpendiculars^ as explained and illustrated in (112 
to 117) all of which is based on (110). 

120. The only other derivative method, which need be mentioned, 
is one which is applicable to bodies bounded by straight lines, which 
are arranged in parallel groups, as in a square prism. In this case, 
the lines of the object itself may be put in perspective by (116). 
The intersections of their perspectives will then be the perspectives 
of the corners of the object. 

Derivative methods, exclusively, are generally used in connection 
with two planes, only, of projection. 

121. We will now close this chapter with three fundamental illus- 
trative examples, showing, j^rs^, how to find the perspective of any 
line whatever by its vanishing point and point of intersection with 
the perspective plane ; second^ how to find the perspective of any 
object by the method of diagonals and perpendiculars ; and, thirds 
how to find the perspective of a plane sided object by finding the 
vanishing and intersection points of its own edges. 

Example 3. — To find the Perspective of a Straight Line 
in any position, oblique to both planes of projection, by 
»ts vanishing point and intersection -with the perspective 
lane. 

Figs. 45 and 46. To familiarize the student more fully with this 
problem, and so to render the conception of positions in space, cor- 
responding to given projections, more easy, two different lines have 
been taken in the above figures, while for more ready comparison, 
like points are lettered with the same letters in both figures. 
Accordingly, ah-a'b\ in both cases, is a line behind the perspective 
plane, as usual. Its extremity aa' is at the distance ac behind the 
perspective plane^ and height, a'c above the horizontal plane. J, 
being in the ground line, is the horizontal projection of that point 



GENERAL PRINCIPLES AND ILLUSTRATIONS. 



88 



of the given line, which is in the vertical plane, that is, the point 
h'. That is ah-a'h' pierces the vertical, or perspective plane at V 
which is therefore one point of the perspective of this line (116). 





^^ 


d' 


• 


a ^^-^y^ 




..E' 






^ I 






• ' ^^ 












^^ ^-."^^^ 






• ^^^^ 








:a' \ 
















vl^f^^ 1 


\ \ 




. \V ! 


\c \ 


I 








s 1 






\ 1 






\ ; 






\ 1 






\ 1 






N ; 






N 1 






N 1 






\ : 






X 1 






% 1 






N 






N 1 






\ 1 






\ , 






^-ii: 







Examples for Practice. 

1. In Figs. 45, 46, find 
the perspectives of one or 
more other lines parallel to 
db-a'b'. 

3. Let the given lines be 
to the left of EE' (Figs. 45, 
46). 

3. Let them be any three 
horizontal lines, other than 
diagonals or perpendicu- 
lars. 

4. Find the perspectives 
of any three points, by 
means of any two auxiliary 
lines through each. 



Fig. 45. 




Again : Eu-E't;', parallel to db-a'h\ is the parallel visual ray 
from the infinitely distant point o^ ah-a'h' (108). Hence v', where 
this ray pierces the perspective plane, is the perspective of that infi- 
nitely remote point. That is, v' is the vanishing point (106) oi ah- 
a'h'. Hence by (116) v'h' is the perspective oi ah-a'h'. 

Hemarks. a. — As v' is the perspective of an infinitely distant 



84 



LINEAR PERSPECTIVE. 



point on ab-a'h\ v'b' is the perspective oidb-a'h' produced to aft 
infinite length, from hh*^ back from the perspective plane. 

h. As a further exercise, let the student take lines in othei 
positions. Thus in Fig. 45 let the given line have such a position 
that a'h' shall be its horizontal, and ah its vertical projection, and 
then find its perspective, as before. 

Example 4. — To construct the Perspective of a Tcwer 
and Spire, by diagonals and perpendiculars. 



,-fe5t 




Fig. 47. 

Fig. 47, let PBE be the plan of the tower, and FGN of the spire 
whose vertex is A. Let LL be the ground line, taken through the 



?! 



<3ENERAL PRINCIPLES AND ILLUSTRATIONS. 8f 

corner B, which indicates the real position of the perspective plane; 
L'L' indicates the position of the perspective plane after translation 
forward. CD is the horizon, and C the centi-e of the picture. The 
vanisliing point of diagonals is assumed on CD, at the left of C, and, 
in this case, beyond the limits of the picture. 

[This preliminary explanation is substantially common to most of 
the following problems, and is therefore to be understood though 
not repeated.] 

Since the vertical edge at B is in the perspective plane, it is its 
own perspective, hence its vertical projection, B'C, which shows 
the true height of the tower, is also its perspective. To find the 
perspective of either of the other visible vertical edges, as the one 
at E, draw the diagonal E5 from the corner, E, of the base of the 
tower. E'6' is the vertical projection of this diagonal, since it is in 
the horizontal plane ; it pierces the perspective plane at b\ and J'D 
(D meaning the vanishing point of diagonals, not shown) is its 
perspective. Em, the perpendicular from the same point E, pierces 
the perspective plane at E', and E'C is its perspective. Hence e, 
the intersection of ^'D and E'C, is the perspective of E, considered 
as in the lower base of the tower, p is found from P in a precisely 
similar manner. Then draw joB' and B'e. 

To find any point, as A, of the top of the tower. E5, considered 
as the diagonal from the point E in the top of the tower, pierces the 
perspective plane at the true height of the tower, that is at c', in 
the horizontal line through C, since diagonals are always horizontal 
lines (112). Then c'D is the perspective of this diagonal. The 
perspective of the perpendicular from the upper point, E, is not 
needed, since the perspective of the vertical edge at E is known to 
be a vertical line through e. Hence A, the intersection of eA, drawn 
perpendicular to the ground line, with c'D, is the perspective of 
the top corner of the tower at E. q is found in the same manner 
Then draw qC and C'A. 

To find any point, as/*, in the perspective of the base of the spire 
/ is the perspective of F, and the plane of the base of the spire ia 
the same as that of the top of the tower ; hence the diagonal, Yd., 
and the perpendicular, F^, pierce the perspective plane in the hori- 
zontal line through C, at d' and F', respectively. Then dT> is the 
perspective of Yd, and F'C, that of Yg ; and/, the intersection of 
these perspectives, is the perspective of F. The top of the towei 
being above CD, the level of the eye, the base of the spire is invi- 
sible. The perspectives of G and N are foimd in the manner just 
described. 



86 LINEAR PERSPECTIVE. 

Finally, the height of A'm' above the ground line, represents the 
height of the top of the spire. Then the diagonal, Am, and per- 
pendicular, Aw, pierce the perspective plane at m! and A'. m'D is 
the perspective of Am, and A'C, that of Kn. Hence a is the per 
spective of A. Now join a with/*, and the other points of the base 
of the spire, limiting the lines thus drawn by qQ>' and C'A, and the 
required perspective will be complete. 

122. Various miscellaneous points, which naturally arise in the 
mind of a beginner, are most conveniently disposed of here, after 
the progress thus far made with primitive and derivative methods. 
They are therefore discussed in the following — 

Memarks. — a. The statements in (101) can now be made more 
intelligible. First. Derivative methods abridge the labor of con- 
struction: first : — through the partial omission of the projections^^ 
as seen in the above example, where the vertical projection was not 
required, because the auxiliary lines, being horizontal, will always 
pierce the perspective plane at the height of the points from which 
they are drawn, and these heights can always be indicated by set- 
ting them off, as b'c' was, equal to B'C, the real known height of 
the top of. the tower ; second : — by the provision of common van- 
ishing points for all parallel lines, so that, 1°, — only one other 
point in each indefinite line, need be found ; and 2", — so that any 
particular point, as A, on such a line, can be found by a single auxi- 
liary, as c'D. 

Second. Derivative w^ethods also conduce to accuracy / first : — 
by providing against errors arising from very acute intersections 
in the lines of construction. See Part I., Fig. 19, where, though 
the intersection at A is well defined, that at F, and especially at the 
perspectives of c" and d" (not shown) are not. Whenever, there- 
fore, another method fails to give well-defined intervsections, that of 
diagonals and perpendiculars will be generally found available. 
Second : by the provision against distortion of apparent propor- 
tions, which is afforded by vanishing points. It is a matter of 
familiar experience, that all receding parallels in the same group 
appear to vanish at the same point, and in a drawing, where 
vanishing points are employed, their perspectives will likewise 
vanish. But if no vanishing points are used, so that the perspec- 
tive of each line of a parallel set is found h^dependently of th€ 
others, by finding two points in it, it may happen that these per- 
spectives, if produced, will not meet at one point. Erroi's in the 
true relative direction of perspectives are far more offensive to the 
eye than the less obvious errors in the absolute place of single points. 




GENERAL PKINCIPLES AND ILLUSTRATIONS. 87 

b. The disuse of vertical projections which the method of diago- 
nals and perpendiculars allows, is another advantage of that method 
over those in which the auxiliary lines might not be horizontal. 

c. The question naturally occurs to a practical inquirer, " how 
shall I represent an object of given dhnenslons, \'iewed from a 
given distance^ and in a given direction^ See Fig. 47. If in 
practice the distances from P and E to the observer be measured, 
the exact relative position of the tower and the observer will be 
known, and so can be laid down on paper. This done, we can, 
from a given position, look straight forward towards the centre oi 
an object, as shown by the line CA in the figure, or we can turn 
and look towards the right or left of the centre so as to see the 
object partially by a sidewise glance of the eye. 

The clearest view is obtained in the former case, but in any case 
the perspective plane is supposed to be perpendicular to the direc- 
tion of vision. Thus, if the spectator at E, Fig. 48, observe O, 

among other things, while looking in the 
direction Ee, the perspective plane PQ 
\^ — ■ ^B should be perpendicular to Eey simply 

\ because this is its simplest and most natu- 

\ ral position. 

\ This being understood, make eD = eE, 

'^ to find the vanishing point of diagonals 

Fig. 48. (H^) ^^\ i^ Fig. 47, lay off from w, on 

AnQ produced, a distance equal to CD, to 

find the position of the observer, or horizontal projection of the 

point of sight, often called the station point (l IS). 

d. But, farther, in representing large objects truly, all these 
dimensions and distances, just spoken of, must be reduced uniformly, 
so as to be shown at all, and in true proportion, on paper. In 
other words they must be drawn to a scale. 

For example, let it be supposed that in Fig. 47, all given parts 
are to be shown on a uniform scale of five feet to an inch, i.e. Jive 
feet on the real object, to one inch on the drawing. On any 
straight line as XY, lay off two or more inches, and divide each 
inch, as shown, into five equal parts. Each of these parts will 
therefore represent one foot, and hence, in connexion with the 
drawing, may he called one foot. Let the left hand one of these 
feet be subdivided into twelfths (fourths only are shoAvn) which 
will be inches. Any other scale is made in a similar manner. 
Having such a scale, its zero point is at the right hand end of the 
divided foot. If then, the tower is 5 ft. 9 ins. square, as at BI* and 



88 LINEAR PERSPECTIVE. 

BE, extend the dividers from the point marked five to the 3 inch 
point between 6 and 12, which will be five feet nine inches, on a 
scale of five feet to one inch. So, if the object be 11 ft. 6 ins. liigh, 
make the line K'm! at this perpendicular distance from the ground 
line LX'. 

It thus nppears that in using any scale, thus constructed and 
numbered, no calculations need be made, since we take up in th 
dividers the same number of scale feet and inches, that there are 
of real feet and inches in any given line to be represented. The 
question of JRem. c is thus fully answered. 

e. It is a familiar fact that the apparent size of an object 
decreases with its increased distance from the eye, but the term 
apparent size is really a little ambiguous, on account of the interfer- 
ence of knowledge with sense impressions. Thus, when I see a 
whole house through one window pane, I perceive that the appa- 
rent size of the house is less than that of the pane, and it is so 
because the image of the house on the retina of the eye, which is 
what determines its real apparent size to simple sense, is less than 
that of the pane. But I ^now that the house is much larger than 
many panes, and this knowledge is so far controlling, that the sight 
of the house affords a mental impression of an object much larger 
than the pane, though the merely sense impression is, that it is 
smaller. 

In relation to the distinction here explained, a completely natural 
artist is one, who sees things, only, and just as, his sense of sight 
sees, without any interference from thought or knowledge of real 
relative sizes ; and who draws objects just as his eye sees them. 

Such a one will spontaneously conform to the principles of per 
spective, which, in relation to him, will only be the natural history 
of his natural performances. In proportion, however, as knowledge 
of the real sizes of objects warps the judgment, as to their real 
apparent size to the eye alone, does a scientific knowledge and 
practice of perspective become necessary as a guard against errors 
in drawing. 

f. According to (14-16) and all the preceding constructions, a 
perspective drawing should be viewed from the precise point from 
which the object represented is supposed to be viewed. Thus, Fig. 
47 should be viewed by the eye placed in a perpendicular to the 
paper at E, and five inches ( — CD) from that point. The per 
spective will then make identically the same image on the retina 
that would be made by the original object in its full size, and 25 ft. 
(the distance by scale) from yi, on the perpendicular AwC. 



GENERAL PRINCIPLES AND ILLUSTRATIONS. 89 

In a picture, properly so called, where the sensible effect is 
greatly assisted by shade and color, if it be viewed through a tube, 
so as to exclude the surrounding objects which warp the judgment 
when compared with the small size of the picture, the illusion may 
be made complete, by abandoning the mind to the picture exclo 
sively, and w^e really seem to look up through extended valleys, 
winding among great hills, and overhung by a real far distant sky. 

g. The principal exception to this rule for the position of the eye, 
is in viewing decorative wall paintings of interiors, which may be 
painted as if seen from a great distance, or otherwise modified so as 
not to be offensively distorted to beholders in any ordinary position 
within the building. 

h. In connexion wdth oblique vision of an object, as mentioned 
ill (c), the question occurs, "to what extent is such vision admissi- 
ble.'' In other words, what is the practical limit of the visual angle. 
We can examine objects with the greatest minuteness only a point 
at a time or in the line of but a single visual ray at a time. On the 
other hand, we can be conscious of the existence of objects within 
a range of 180°, either vertically or horizontally. Where, now, 
between these limits, is the greatest visual angle which will allow 
of a clear and pleasing general effect ? It is usually supposed to 
vary from 45° to 60°. 

Accordingly, in Fig. 47, by laying off five inches, = CD, in front 
of w, to obtain the station point (113) (115), and from this point 
drawing lines to P and E, it will appear that a small visual angle is 
formed. Hence when Fig. 47 is viewed, as directed in (/"), it wall 
be very clearly seen. 

i. This clear view is also due to looking directly, in the line CA, 
at the centre of the object. Thus Fig. 47 is much more satisfactory 
than Figs. 16 and 17, Part I., where the eye is placed considerably 
to one side of the given object, partly to avoid the confounding of 
plans and perspectives, and partly to avoid the very acute intersec- 
tions of lines of construction that would have occurred had the 
point EE' been placed directly in front of the objects. 

The last consideration points to another disadvantage of the 
method of visual rays, especially as employed in connection with 
two planes only. 

Example 5. — ^To find the Perspective of a Cross and 
Pedestal. 

This problem is chosen as one embracing numerous lines arrangsd 
in parallel sets. 

In Fig. 49, let ABD be the plan of the pedestal, EFG, ai 



90 



LINEAR PERSPECTIVE. 



the horizontal arm of the cross, and HIK, that of its vertical 



arm. 



LL is the ground line which indicates the fii'st, and L'L' the one 
which indicates the second position of the perspective plane. W 
is the level of the eye, and therefore by (105) contains the vanishing 
points of all the horizontal lines of the object. S is the station 
point (113) taken in a perpendicular to the perspective planr* 
through the centre of the object (122c). 




Fig. 49. 



Then, drawing SL, parallel to AB, and LV perpendicular to LI^ 
we find V, the vanishing point of all lines in the direction of AB. 
In a similar manner Y' is found. Other points in the indefinite 
perspectives of the horizontal lines, are where those lines pierce the 
perspective plane. 

Accordingly, as shown by the figure, and (I22a) and assuming 
ca as the height of the pedestal, MB, produced, meets the perspec- 
tive plane at m\ where nim!' =z ac / and BA meets the perspective 
plane at a. Then m'V is the indefinite perspective of BM, and 
aV, that of AB. Hence J, their intersection, is the perspective of 
B. From J, draw be, perpendicular to the ground line and limited 



GENERAL rRINCIPLES AND ILLUSTRATIONS. 91 

by cY^ and one face of the pedestal will be represented. Tlie con- 
struction of its other visible surfaces is similar to the foregoing, as 
is seen in the figure. 

To find the foot of the vertical arm. Kp and Kg- pierce the per- 
spective plane at p' and q\ at heights equal to ac. p'V and q'V 
are their perspectives, which intersect at the perspective of K in 
the foot of this arm. Other like points are similarly found. Ou 
either of the same perpendiculars through p and q, set off the height, 
as at r, of the whole cross and pedestal, then rV, the perspective of 
pK in the level of the head of the arm, will meet the right hand 
edge at k, the perspective of the top point, K. 

To find points in the perspective of the short arm EG. Produce 
its horizontal edges to the ground line, as at q and n. Set off, as 
before, the heights of these points, as at q" and n' in the plane of the 
to]) of this arm. Then ^, for example, the intersection of ^'"V and 
n'Y', will be the perspective of G. Other jDoints may be similarly 
found. 

Remarhs. — a. It is evident from this problem that the method 
by horizontal lines of the object, when such exist, is as convenient 
in resjject to avoiding the necessity of full vertical projection:^, as is 
the method of diagonals and perpendiculars. 

b. The intersection at 5, for example, is very acute, but would 
have been less so had a perpendicular, through B, been used, 
together with aY, Thus, by an adaptation of auxiliary lines to the 
conditions of each point, we can obtain the perspective of each by a 
well defined intersection. 

c. The passing of SL through e, and of SY through (7, are merely 
accidental coincidences, which are always liable to occur, but which 
never need perplex the draftsman if he will retain a clear view of 
first princii^les, and keep in mind what each line of the figure really 
means. 

Examples for Practice. 

1. In Fig. 47, let there be passages through the tower, two feet nine 
inches wide and four feet high, on the supposed scale (p. 87, Rem. d), from 
the sides BE and BP, and replace the spire by a roof with its gable on the 
face BE. 

2. In Fig. 49, replace the cross by an octagonal prism ; let there be a rec- 
tangular recess in each side of the base ; and draw the whole on a much larger 
scale. 



9« LINEAR PEKSPECTIVE. 



CHAPTER n. 

PERSPECTIVES OF SHADOWS. 

123. Returning to Fig. 35, Part I., where BR represents a ray of 
light and AR, its horizontal projection (96), it is evident that R, the 
shadow of B on the horizontal plane, is the intersection of the ray 
through B, with its horizontal projection. 

It follows, now, that the perspective of R will he the intersection 
of the perspective of the ray with the perspective of its horizontal 
projection (117). 

If, then, we can find these latter lines, we can find the perspective 
of the shadow directly, or without finding its projections. 

124. But rays of light, proceeding, as usually supposed, from tho 
sun, are parallel ; hence their vanishing point, like that of other 
parallels, is found by determining where a parallel ray of light 
through the point of sight pierces the perspective plane (108). Also, 
rays being parallel, their projections on either plane will be parallel 
(53). Their vertical projections, being lines in the perspective 
plane, will be their own perspectives; and their horizontal projec- 
tions, being horizontal lines, will have a vanishing point in the hori- 
zon (105). Hence the perspectives of rays, and of their horizontal 
projections, can be found ; and therefore perspectives of' shadows on 
the horizontal plane, can be found directly, or w^ithout previously 
finding the projections of those shadows, as in Part I. 

The principle just established, combined with other general ele- 
mentary principles already employed, will also, as will soon appear, 
3nable us to find, directly, all ordinary shadows. 

We proceed next to illustrate the principles just explained, in 
some elementary constructions. 

Example 6. — To find the vanishing- point of given Rays, 
and of their Horizontal Projections. 

Let R and R', Fig. 50, be the two projections of a ray of light, 
which lies in front of the perspective plane, GL being the ground 
line. Also, let EE' be the point of sight. Then EA, parallel to R, 
and E'V, parallel to R', are the projections of a visual ray parallel 
to the raj/s of light. This visual ray meets the perspective plane in 




Cr 



R 



JU 



/ 
/ 



E 



PEKSPECTITES OF SHADOWS. 93 

the point whose horizontal projection is A, and whose vertical pro- 
jection, or the point itself, is V (58). Hence V is the vanishing 
point of all rays of light parallel to R-R'. 

Again, remembering that the 
vertical projection of a hori- 

lE JS.. zontal line, oblique to the ver- 

I ticai plane, is a line parallel to 

''It t^^ gromid line (51) E/i, again, 

parallel to R, and E'H, paral- 
lel to GL, are the projections 
of a visual ray, parallel to the 
horizontal projections of the 
rays of light. This visual ray 
pierces the vertical or perspec- 
FiG. 50. tive plane at H, which is there- 

fore the vanishing point of 
horizontal projections ofrays^ and is in the horizon E'H. 

125. Having completed Fig. 50, observe that H and Y, are, by 
construction, necessarily in the same perpendicular to the ground 
line. This follows from the fact that a ray, and its horizontal pro- 
jection, are in the same vertical plane (81), and therefore the visual 
rays, as EA-E'Y, and E/i-E'H, parallel to them, are in a vertical 
plane ; while the vertical trace of such a plane, in w-hich these visual 
rays pierce the vertical plane of projection, is a line perpendicular 
to the ground line (79). 

126. In general, if a ray, or any other line, be contained in a cer- 
tain plane, it must pierce any surface in the intersection of the plane 
with that surface ; that is, in the trace of the plane upon that sur- 
face. 

127. When a particular direction of the light is given., as at 
R-R', H and V must he constructed by (Ex. 6), but if, as is usual in 
general problems, its direction is not given., H and V may he 
assumed., agreeably to (125). 

Example 7. — To find the Perspective of the Shadow ot 
any Vertical Line upon the Horizontal Plane. 

Let B'L, Fig. 51, be the ground line ; which, in figures of so few 
lines as Figs. 50 and 51, and similar ones, need not be translated. 
Let A be the horizontal, and A'B' the vertical projection of the 
given vertical line. Its perspective, ah., is found by visual rays, a8 
AE — A'E', as explained in Paet L, EE' being the point of sight. 

Then, by (125) assume V and H, as the vanishing points of rays 
and horizontal projections of rays, respectively. Then aY is tho 



9i 



LINEAR PERSPECTIVE. 



perspective of the ray of light through the point whose ])erspeo 
tive is a, that is through A A', and bH is the perspective ol' the 



A' 




H 






V 


V ^ 1 -» 


T- 








v^ 





Fig. 51. 



horizontal projection of the same ray. Hence by (12n) 11 is the 
perspective of the shadow of a upon the horizontal plane ; and h\\ 
is the perspective shadow of ah on the same plane. 

Remarks. — a. Remembering that when the rays are parallel, as 
here supposed, they, and their horizontal projections, will, each, 
have a common vanishing point, it is evident, that if the shadows 
of a number of vertical lines, like a^, be found on the same hori- 
zontal plane, they will all converge to the point H. The student 
should make this construction. 

h. When the vanishing point V is below the horizon, and the 
light proceeds as indicated by the arrow in Fig. 51, it shows that 
the light proceeds from above and behind the left shoulder. If V 
nere above H, it would indicate that the light proceeded from 
above and in front of the right shoulder ; and the shadows would 
fall towards the observer and to the left, as will readily appear on 
making the construction. 

c. If the source of light were a near point, as a candle, the per- 
spective of this point and of its horizontal projection must be 
found. Lines irom the former to points in the perspective object 
will be the perspectives of rays ; and lines from the latter point 
to the perspectives of horizontal projections of the same jioints of 
the object, will be the perspectives of horizontal projections of rays 



PERSPECTIVES OF SHADOWS. 



95 



The construction is left as an exercise. 

Example 7«.— To find Shadows on Side Walls. 

Let ab, Fig, 51a, a vertical, and ac, an oblique rod, meet the 
floor, one at h, the other at c. GL being the ground line ; E'H, 
the horizon ; E', the 

point of sight ; and N 

V and H, the van- 
ishing points of 
rays, and of their 
horizontal projec- 
tions, let MNOP be 
the perspective of a 
wall perpendicular 
to the perspective 
plane on MN ; and 
therefore, having E' 
as the vanishing 
point of its top and 
bottom edges, NP 
and MO; and, hence 
(105-107), having 




Fig. 51a. 



jSTow : 

of a vertical 



E'V, perpendicular to GL, for its vanishing lino. 

First. JH is the perspective horizontal trace 
plane of rays containing ab ; then rt, parallel to ab, is the per- 
spective trace of this plane on MNOP. Hence the ray «V, lim- 
iting rt at e, gives br, the shadow of ab on the floor, and re that 
on the wall. 

Again : aV, meeting bH at n, gives 7i the shadow of a on the 
floor plane ; and, thence, en the shadow of ac on the floor ; of 
which cs is real, and se the shadow of ac on the wall. 

Second. Drawing bb' and aa' parallel to GL ; b'a\ parallel to 
ab, limits aa' at a', the projection of a on MNOP. Also, project- 
ing V at y, on E'V, gives V as the vanishing point of projec- 
tions of rays on MNOP (105). Hence, e is also found as the 
intersection of the perspective ray aY with a'Y', the perspective 
of the projection of that ray on MNOP, by a simple extension 
of the principle of (123). 



03 



LINEAR PERSPECTIVE. 



CHAPTER IIL 



MISCELLANEOUS PROBLEMS. 



128. The following problems are added, not to illustrate any ne^w 
principles, but to familiarize the student more fully with the appli- 
cation of those already explained, to practical problems. 

Premising that the drawing of exterior and interior views of 
buildings, with their accompaniments ; arcades, pavements, and 
furniture, is perhaps the chief exact application of perspective, this 
chapter is occupied with examples of this character, the execution 
of which will enable the draftsman to proceed with the perspective 
drawing of Avenues, Bridges, Machines, etc., and with the correct 
additions of features of natural scenery to the geometrical portions 
of his drawings. 

Example 8. — To find the Perspective of a Pavement of 
Squares, -whose sides are parallel to the ground line. 

Let GB, Fig. 52, be the ground line, DC the horizontal line, or 
horizon, and AGK a group of twelve squares, lying in the horizon- 
tal plane, and with one side, GK, taken as the ground line. 





M 




A 




I, 


D 








c '\. 


E 




"*-^-~-_. 






/i 


'\"*^ 


\^ 






■ — -^^,,^^ 




/' i 


\ ^\ 


*x 






■**.. 




t / 


* *^ 


^^ 








""••^ 


f 1 




* 








~~--. 


./ ,' 




^x 


P 








' 


a. 


^ 


^^N^ 






^- — — -■■■ 


^ 




^^f*.* 


^ 


__ 


G 


^ — 


F 


- \ 


\ '"----nB 



S. L 

Fig. 52. 



jb: 



Operating by the method of diagonals and perpendiculars, let C be 
the centre of the picture (113) and D the vanishing point of diago- 
nals. C is the vanishing point of perpendiculars (112) and these 
perpendiculars, as LK, AH, etc., pierce the perspective plane at K, 
H, etc., hence (116) KC, HC, etc., are their perspectives. AB if? 



MISCELLANEOUS PROBLEMS. 



97 



the diagonal from A and BD is its perspective. Therefore a is the 
perspective of A. Likewise e is the perspective of E, and/", of F. 
Drawing parallels to GK, through a, e, and/", (ill) they will inter 




i— V 




^lO. 53. 

fcect the perspectives of the perpendiculars, so as to compIet)e the 
required perspective QcmTK. 



9S LINEAR PERSPECTIVE. 

Example 9. — To find the Perspective of a Pavement of 
Hexagons, -whose sides make angles of 30° and 90° with 
the ground line. 

Let HE be the ground line, Fig. 53. Construct an equilateral 
triangle, as ABD, with one of its sides perpendicular to the ground 
line. Divide either of its sides, as AB, into any convenient nnm 
her of equal parts. Through each of the points of division, as Q, 
draw indefinite lines, as QN and QP, parallel to the remaining 
sides, DB and DA, of the triangle. 

Portions of these lines, together with pei-pendiculai-s, as OP, 
joining the proper intersections, which will be obvious on inspection, 
will form a group of regular hexagons. These may be limited at 
pleasure, as by the rectangle HEFG. 

[N'ow let LL be the ground line, after translation, CV the hori- 
zontal line, C the centre of the picture, and S the station point, 
taken in this case, for variety, at one side of the middle of the 
figure. 

The sides of the hexagons, forming parallel groups, are taken 
as lines of construction. Their vanishing points — beyond the limits 
of the figure — are found by drawing lines at S (partly shown) 
parallel to HK and IG, till they meet HE. From the latter points, 
perpendiculars to CV produced, will meet CV in the vanishing 
points of HK and IG and of all linos parallel to them. The re- 
maining lines of the hexagons are perpendiculars, and C is their 
vanishing point. 

Observing that the plans and perspectives of the same points 
have the same letters, the remainder of the construction needs no 
furthei explanation. 

Jiemark. If, in Ex. 8, the squares had been placed with their 
sides making angles of 45° with the ground line, those sides would 
all have been diagonals, instead of parallels and perpendiculars. 

In the last example, if AB had been taken in the ground line, 
he sides of the hexagons would have made angles of 60° with 
he ground line, except those which would have been parallel to it. 
Hence, SR remaining the same, the vanishing points of the inclin- 
ed sides would have been nearer to C. The student should re- 
construct these examples under these new conditions. 
Example 10. — ^To find the Perspective of an Interior. 
Preliminary explanation. According to (122A) a person stand- 
ing against one wall of a room, can be conscious of the entire in- 
terior, though the whole cannot be distinctly recognized. If, then, 
Fig. 54, a person stand at E, seeing clearly everything within the 



MISCELLANEOUS TROBLEMS. 



99 




Fig. 54. 



visual angle AEB, only the portion of the room be- 
yOnd AB can be represented in a picture. Hence, 
if a larger portion, or the whole of the interior is 
to be represented, the near wall A"B" must be sup 
posed to be removed, so that E', or E", may be the 
position of the observer, from which all beyond A'B', 
or A"B", will be visible. 

Construction of Fig. 55. In this example, let 

the near wall be removed, and let the whole interior 

be seen under a visual angle of 45". ABGL is 

the plan of the room, with an elliptically arched 

passage, of the width EF, opening out of it on the right, and with 

a door, HK, in the left wall. 

Let the observer stand opposite the point X, at one third the 
width of the room from G. We have then to construct S, the ver- 
tex of an angle of 45°, whose base is GL, and placed opposite to 
X. Draw GT and LT, each making an angle of 45*^ with GL. 
Draw, at X, a perpendicular to GL, and with T as a centre and 
TG as a radius, describe a small arc, intersecting this perpendicular 
at S, which will be the station point as required. 

Now let GT<' be the ground line, indicating the second position of 
the perspective plane, and let CD be the horizontal line. This 
line must, if the observer is supposed to stand on the floor, be 
d}io\it Jive feet above GL', on the same scale on which the plan, 
AG, is drawn. Note that C is in the perpendicular XS, produced 
to meet CD. 

Observing, now, that a diagonal from A will meet LG at a dis- 
tance to the left of L, equal to LA, and so for other points, the 
diagonals themselves need not be drawn. Thus, make CD _ SX 
(115) and D will be a vanishing point of diagonals. Then make 
L'A =LA and A'D will be the perspective of the diagonal from 
\. The perspective of the perpendicular LA is L'C, hence a is the 
perspective of the right hand back corner. A, of the floor. Draw 
ab^ parallel to L'G', till it meets G'C, the perspective of GB, and 
L'G' ah will be the perspective of the floor. 

The front wall of the room, GL, being taken as the perspective 
plane, the intersection of the room with that plane will be its own 
perspective, m full size and real form. Hence make L'L" and G'G" 
equal to the height of the walls ; and, supposing the ceiling to be 
semicircular, describe a semicircle on L"G" as a diameter. 

As an example of a simple cornice, in perspective, make the 
small rectangles at L" and G", as sections of it in the perspective 



100 



LINEAR PEPwSPECTIVB. 




ft' FI Q aO' M' E'li' 

Fig. 55. 



MISCELLANEOUS PROBLEMS. 101 

plane. Then draw its edges towards C, limiting it by a horizontal 
and vertical line where its lower back edge, on each wall, meets 
the vertical lines from a and 5. 

QC is the perspective of a perpendicular through the centre of 
the floor. Hence qs is the perspective of the centre line of the fur- 
ther wall. Where qs meets a's, a' being the intersection of aa' 
nd L"C, is the centre of the semicircular boundary of the further 
end of the ceiling ; which is a semicircle in perspective, because it 
is parallel to the perspective plane. If there be a round topped 
window in the centre of the further w^all, lay off its half width, 
QR=QI, each side of the middle point Q, draw RC and IC, and per- 
pendiculars to ah as at r. Then make Uv equal to the height of 
the base above the floor, draw vC^ and v'r' paralled to a J, and 
the semicircular top, withs as a centre, to have it concentric v\ ith the 
ceiling. This will complete the outline of the window. 

To draw the opening HK (which is very wide in order to show 
the construction more plainly). Draw HS and KS, horizontal pro- 
jections of visual rays, or horizontal traces oi vertical visual planes^ 
through the sides of the opening. Then the intersections of these 
planes with the perspective plane, at hw and Jc, drawn from H' and 
K', will be the perspectives of the vertical doorway lines at H and 
K. Make G'W equal to the height of the door, and WC will 
limit the inside of the top of the door. Next draw the edges in 
the thickness of the doorway as kl^ parallel to G'L', the vertical 
line It^ and from t the line towards C, which completes the doorway. 

To draw the archway EF. Make L'E'=LE, and L'F'=LF. 
Draw ED and F'D, which will give e and/*, the perspectives of E 
and F. Make L'O" equal to the height of the vertical portion of 
the archway, and limit the vertical lines at e and /*, by 0"C. To 
find the perspective of the highest point, draw the semi-ellipse, 
FPE, representing the elliptical top of the arch as revolved I'ound 
EF, till parallel to the horizontal plane. Lines, as OP, in this 
Bemi-eUipse, are called ordinates. Take the longest ordinate, OP, 
set it off at O'T" and draw P"C. Make L'0'=IX), draw O'D, 
and op^ then p will be the perspective of O, that is of P. Any 
point in the ellipse may be similarly found. Thus, take MN" any 
where, and parallel to OP. Make 0"]S['=Ml!^, and draw N'C. 
Also make L'M'==LM, draw M'D and mn^ then w, the intersection 
of onn with N'C, will be the perspective of IST. After finding one 
or two more points in like manner, the perspective ellipse, fpne' 
can be sketched. The horizontal line at /"will then complete the 
archway, and the whole figure. 



102 



LINEAR PERSPECTIVE. 



Example 11. — To find the Perspectives of the Shadows 
in an Interior. 

In order not to confuse figure 55, the constructions of the re- 
quired shadows are made on the following enlarged copies of 
detached portions. 

First ; to find the shadow of the edges of the doorway, Fig. 56. 
Supposing no particular direction of the light to be given, assume 
H as the vanishing point of horizontal projections of rays, and R, 




Fig. 56. 



as the vanishing point of rays. It is readily apparent on considera 
tion, that ka^ tc and cd are those edges of the doorway, parts of 
which, at least, will cast shadows. kR is the perspective of the 
horizontal projection of all rays through the vertical edge lea. That 
is, it is the horizontal trace of a vertical plane of rays (99) througli 
Jca. It is therefore the shadow of Jca on the floor, as far as m, 
where it meets the further wall ABD. Thence, this plane being 
vertical, its trace, and shadow of ka on ABD, is vertical, as seen at 
mE. 

To find E, consider that ct will partly cast a shadow on the 
surface atk. This surface being parallel to the perspective plane, 
and ct perpendicular to it, the shadow of ct on atk will be parallel 



MISCELLANEOUS PKOBLEMS. 



108 



to the vertical projection, CR, of a ray of light (97), and will begin 
at t^ where ct meets atk. Hence ife, parallel to CR (111) is the 
shadow of ct on atJc. Therefore e is the highest point of Jca that 
can cast a shadow. Hence draw the ray eR, and E, its intersection 
with wiE, will be the limit of the shadow of ka. The remainder 
of the construction is now evident from the figure. 

Since light streams through the doorway, the area within th 
shadow of its edges is light, as indicated by the partial shade line 
of the figure. 

Second. To find the shadow in the archway, Fig. 57. This 
shadow is in four parts ; first, the shadow of the edge ee' upon the 
floor ; second, thnt of the same edge on the wall Qnff ; third, that 
of the curve epf on the same wall ; and fourth, that of the same 
curve upon the cylindrical surface of the archway, above the hori- 
zontal plane through e' and/". 

Let H be the vanishing point of horizontal projections of rays, 
and R, the vanishing point of rays. Then eH is the perspective of 



U.^d/ 




F 



O 



Fig. 57. 
the horizontal trace of a vertical plane of rays through ee\ and eG 
is the shadow of this edge on the floor. By drawing the ray RG, 
and producing it to g, we learn that eg is the precise portion of ee! 
which casts a shadow on the floor. From G, the shadow of ge' ia 



104 LINEAR PERSPECTIVii:. 

the vertical line GE, limited at E by the ray e'R. Above E, the 
shadow is cast by the arch curve, and is found as follows. Assume 
any point, q ^ and draw the vertical line qq^ which is the trace of a 
vertical plane of rays through ^', upon the side of the room. Then 
g'H is the perspective of the trace of this plane upon the floor, and 
the vertical line from the intersection of g-H with/"G, is its trace on 
the wall of the arch. This plane contains the ray g-'R, which meets 
the vertical line, just named, at Q, which is therefore the shadow oi 
2''. D, the shadow of d' is found in like manner. T, the point of con- 
tact of the ray TR, with the arch curve/^e', is the upper end of the 
shadow, which may be sketched by joining the points ah*eady found. 

In this figure, the shadow on the cylindrical surface of the arch 
is so small, that no points in it have been found, except T. The 
most elementary method of finding points of this shadow, is the 
following indirect one, which is so fully indicated that the student 
will probably find no difficulty in applying it for himself. Assume 
any point as A quite near to /*, on L'a, and draw through it lines 
parallel to yO and ff\ which will represent a plane parallel to the 
wall ^ff» This plane will cut a line from the arch^ parallel to /■&, 
and beginning where the vertical line from h meets the arch curve. 
Next find a few points of shadow on this plane, just as DQE \vas 
found. Then the intersection of this auxiliary shadow with the 
horizontal line cut from the arch, will be a point of shadow on the 
arch (99) and by drawing a ray from R through this point, we can 
find the precise point onfpe' which casts this point of shadow. 

Example 12. — To find the Perspective of a Cabin. 

In this example, a variety of methods will be employed, by way 
of review ; also some special operations, suited to the construction 
of particular points. 

Let ABD, Fig. 58, be the plan of the cabin walls, EF of its roof 
ridge, and H"HI of Its chimney. Let the perspective plane be 
taken at GK, through the corner A, and let G'K' be its ground 
line after translation and revolution into the plane of the paper. 
Let VV be the horizon, C the centre of the picture, and S the 
station point. The perpendicular to the ground line, and contain- 
ing S and C passes through * the centre of the plan (122 i). 

The edge at A, being in the perspective plane, is its own perspec- 
tive, and appears in its real height at aa'. The visual ray BS — • 
L'C pierces the perspective plane at ^, the perspective of the lower 
corner at B. Make L"B' '=««', then BS — B"C is the visual ray 
from the upper corner at B, and h' is the perspective of that corner. 
Draw ah and a'h\ 



MISCELLANEOUS PROBLEMS. 



lOfi 




Fig. 58. 



206 LINEAR PERSPECTIVE. 

The vanishing point of all lines parallel to AB, can now be found 
in either of two ways. In the usual way, it would be found by 
drawing through S a line parallel to AB, till it meets GK, whence 
drop a perpendicular to W (108). Or, produce ah and a'b' till they 
meet W, in the same vanishing point ; which, being out of the 
paper, is indicated by V". 

Likewise find V, the vanishing point of all lines parallel to AD, 
in the usual way, if before finding dd\ or as just explained, if after 
finding dd\ as shown in the figure. Having found the end, add\ 
of the cabin, the intersection, e, of its diagonals ad' and da\ is the 
perspective centre of that end, over which, in the vertical line ee\ 
the peak of the roof is found as follows. Lay ofi* the real height, 
projected from E, at E" ; then E"C, the perspective of a perpen- 
dicular from E, will intersect ee' at e', the perspective of EE". 
N^ow draw a'e' and d'e\ the perspectives of the left end lines of 
the roof. These lines are in the same vertical plane with ad^ hence 
their vanishing points are in the perpendicular, GG', to G'K' and 
through V (125-6). Hence produce a'e' to meet GG' in R, which 
will |be the vanishing point of all lines parallel, in space, to a'e'. 
Also e'd'^ produced to T, makes T the vanishing point of all lines 
parallel to e'd'. To find R and T by the usual process, consider 
that a'E" and D"E" are the vertical projections of AE and DE, 
and then find where lines through the point of sight C, S, and 
parallel to a'E" — AE and D"E" — DE, pierce the perspective plane, 
which will be, as before, at R and T. Next draw e'f to V", and 
h'fio R, which will complete the perspective of the roof. 

To find the perspective of the chimney, and first of its basei 
Produce IH to J and, drawing the visual ray JS, project J' into 
the edge of the roof at j. Then draw jh through V"; or, by ele- 
mentary geometry, draw h'f" parallel to a'e'^ and limit it by e'f 
produced, then divide a'e' and h'f" proportionally at j and f 
and draw jf (For example, if e'J is J of ^'«', then make/"''/"=^ of 
-^''h'). Find h by the visual ray HS, whose intersection with the 
perspective plane at H' is projected into jf at h. Find u in the 
same way from I. To find ^■, set up the full height of the chimney 
top from the ground at ^", projected from I, and draw the perspec 
tive perpendicular I'C to limit the vertical edge ui at ^. Other 
wise: (Ex. 5.) produce the right hand side of the chimney to Y , 
and set up its height, projected from I", at i"^ and limit ui by 
i"Y. Then limit hh' by ih' drawn through Y"\ and draw h'Y, 
Draw AR until the ridge is met, thence a line towards T, limited 
as follows. DrawJV and notej', its intersection with e'd'^ whence 



MISCELLANEOUS PROBLEMS. 107 

draw a line, j'h"^ to V", limiting an edge of the chimney at K\ 
whence draw this edge, which is limited by A'Y. 

In finding the door and window, further special constructions 
will be used, as proposed. 

If lines be drawn, parallel to any line as BK, AB and AK will 
be similarly, that is, proportionally divided, and if AK=AB, these 
similar parts will be equal, and in the same order. Hence make 
aK'=AB, and YJh will be the perspective of KB, and V", its in- 
tersection with W, will be the vanishing point of all parallels to 
KB, (106). Then make aP and aO equal to the distances of the 
two sides of the door from a, that is from A, and draw PV" and 
OV" which will meet ah at jo and o, the perspectives of the sides 
of the threshold. Set off the true height of the door from a on 
aa' and draw a line to V"', which will complete the door by limit- 
ing the verticals 2Xp and o. 

In like manner, a window in the front of the cabin could be put 
in perspective. ^ 

To find the perspective of the end window. According to the 
method just explained, make aG'—^AD (G' accidentally falls on the 
perpendicular RT (Ex. 5. JRem. c.) and let NM be the true relative 
width and place of the window. Draw G'(iV', analogous to K'^, 
also NY' and MV. At n and m draw vertical lines, and having 
made aa!' equal to the height of the window seat, limit them by 
a"V. Make 6«Q equal to the thickness of the cabin wall, and draw 
QV", noting q^ its intersection with db. Then draw qq\ limited 
by a"Y"\ and from q' draw q'Y^ which gives the inner edge of the 
window seat. From the further upper and lower outer corners of 
the window, short lines are seen, which being perpendicular to the 
end wall, are parallel to «5, and therefore vanish at Y'". The 
lower one of these lines is limited by ^'V, and from the point 
thus given, the inner vertical line of the window is drawn, which 
at y limits the upper line to V", and the inner top line which van- 
shes at V. 

The horizontal lines of the fence and sidewalk vanish at V. The 
I op of the fence being in the line VC, it is thus shown to be about 
five feet high. At Z is shown a fi-agment of a cross street, parallel 
to the perspective plane. The real distance of this street from «, 
or A, is equal to the distance from a to the intersection of VZ 
produced (not shown) with aG' produced. The tree in the yard is 
seen, by comparing with the top line of the window, to be about 
twelve feet high. 

129. By comparison of Fig. 58, w^ith any of those in Part I., it 



108 



LINEAR PERSPECTIVE. 



appears, on inspection, that the perspective, as L'C, of aperpendt- 
ciilar, is the same as the vertical projection, L'C, of a visual ray, 
BS — L'C, though the same point. This is evident from the 
definitions of these lines. Each, it will be seen, joins the vertical 
projection of the point through which it passes, with the vertical 
projection of the point of sight, which latter is the centre of the 
icture. 

Example 13. — ^To find the Perspective of the ShadoTW of 
a Chimney on a Roof. 

Fig. 59 is substantially an enlarged copy of a part of the roof of 
Fig. 58, but with the proportions, and the level of the eye, changed, 
merely to bring the construction within the limits of the page. H, 




in the horizontal line, is the vanishing point of projections of rays 
on any horizontal plane (Ex. 6). R, on HR, but not shown in its 
real position, is the vanishing point of rays. 

Now to find the shadow of the edge hW upon the roof. This 



MISCELLANEOUS PROBLEMS. 100 

shadow will be the trace on the roof, of a vertical plane of rays 
through hh! , The line aA, joining the intersections of the edges of 
the chimney with the roof, is horizontal in space, and in the side 
surface of the chimney, within the roof, cc? is a line in a vertical 
plane through the ridge Dc, and is also in the same side of the 
chimney. Hence dY '"^ drawn to V" (see Fig. 58.) is the trace of 
the central vertical plane, through Dc, upon the horizontal plan 
through ah. Now AH is the perspective oi the trace of a plane of 
rays through hh' upon the latter plane ; ew, a vertical line from the 
intersection of AH with d\"\ and meeting the ridge Dc at n, 
is the trace of the same plane of rays on the vertical plane 
through Dc. Hence nh is the trace of the plane of rays 
upon the roof. The ray A'R, in this plane, meets this trace 
at ^, which is therefore the shadow of A'. Hence th is the shadow 
of AA'. A portion of the shadow of h'h is visible, which is found 
as follows. Produce Ac to meet ah in r, which is therefore the in- 
tersection of ab with the front side of the roof, produced. Next, 
produce hn to T in the perpendicular RHT, and T will be the vanish- 
ing point of all traces of vertical planes of rays on the front roof. 
Hence Tr is the perspective trace of a vertical plane of rays 
through ab upon the indefinite plane of the front of the roof. 
Drawing the ray 5R, it gives/" as the shadow oib upon the front 
roof produced. Hence ft is the shadow of hh! on this roof, and 
the portion, s% of this shadow is real, and visible. 

JRemark, In concluding these examples of shadows, and of this 
volume, it may be added, that though few, the student will find 
them so varied, that, by attentively considering them, he will doubt- 
less be able to construct any ordinary shadow, or at least to judge 
more accurately of the appearance of shadows, sketched directly 
or without construction. 

Examples for Practice. 

1. Find the perspective of the circle in the horizontal plane (Fig. 23) by 
the method of diagonals and perpendiculars. 

2. Find the perspective of the vertical cylinder (Fig. 26) in the same 
manner. 

3. Let the given circle be in a vertical plane perpendicular to the perspec- 
tive plane. 

4. Let the axis of the cylinder be horizontal, but oblique to the perspec^ 
tive plane. 

5. Find the perspective of a cone standing on a thin cylindrical base of 
greater diameter than the base of the cone. 



110 LINEAR PERSPECTIVE. 



CHAPTER IV. 

PICTURES, AND AERIAL PERSPECTIVE. 

13'\ For full instructions on the subject of this chapter, those 
who make perspective drawings, primarily for pictorial effect, 
should consult books which treat of perspective as an imitative art 
(27). A few topics only are here treated, principally for the 
information of those who may have occasion to add a few subordi- 
nate items of scenery, &c., to drawings mainly of a geometrical 
character. 

131. Landscape outlines. These, to be sketched in their true 
apparent position and foi'm, must be seen with truly artistic or 
childlike sense, that is, wdthout interference by the knowledge of 
their real position and form. In other words, we simply copy 
what the eye sees, and just as it sees it, and not what the mind 
infers from what the eye sees (122 e). 

132. In proportion as we abandon reliance on simple sense, for 
scientific knowledge, will the unassisted eye fail to serve us per- 
fectly ; and some mechanical guide to it vvill become necessary or 
convenient. 

It would be impracticable, however, to make preliminary plans 
and elevations of broad landscape areas, such as have been employed 
in the preceding geometrical constructions. Hence simpler aids 
are sought. Among these is the use of a pencil and string^ as a 
measure of relative apparent sizes and spaces. Thus, if one end of 
a cord of fixed length be held in the teeth, and a pencil, attached 
to the other end of the cord, be always held perpendicularly to the 
string, the latter being always horizontal, it will be easy to measure 
on the pencil the apparent dimensions of objects, and the distances 
and directions of lines betw:een different prominent points in the 
view to be drawn. After this, the remaining outlines can be 
sketched by the eye alone. 

A more complete guide to the draftsman is a frame of threads. 
Thus, by interposing at a suitable distance, that is, so as to include 
the whole of a proposed view, a rectangular frame, carrying threads 
which cross each other at right angles about an inch apart, the 



PICTURES, AND AERIAL PERSPECTIVE. 



Ill 



actual landscape will be, to the eye, divided into square inches. 
The paper being then divided into similar squares, all that is seen 
in each thread square can be accurately located, by reference to its 
gides, within the corresponding square on the drawing. 

133. This last process simplifies picture drawing, by making the 
whole view consist of the sum of many little pictures, each of 
which is so small, that by fixing the undivided attention of the ey 

pon it, it can be drawn just as it is seen, according to (131). By 
keeping this in mind, and by gradually enlarging the squares, 
either of the frame alone, or the picture alone, or of both, the eye 
may be trained into independence of such guides. 

134. Landscape details. Trees. These, if few, large, and near, 
and in a real field, yard, or street, may have their position and 
height indicated in projection, as in the preceding geometrical 
constructions. Their perspectives will then serve as guides in 
sketching smaller similar objects. Remote trees will be known as 
larger than near ones, if they rise higher above the horizon, while 
standing on the same level. Straight rows of trees may be more- 
accurately drawn, by locating the vanishing point of the line along 
which they are ranged. 

134. Sills will be known by their rising above the horizon ; and 
their relative distances, either by the cutting off of their crest 
lines ; or by their height above the horizon, if of equal heights ; or 
by their dimness of color and shade when finished. 

Thus in this little sketch, the 
mountain stream in ascending, 
flows between the foremost, or 
right hand hill, and the more 
remote left hand one, while the 
dim central peak is evidently 
distant and quite high. 

136. Yalleys^ below the ob- 
server's level, will be known by 
animals, shrubs, &c., in them 



appearing below the horizon. 
Fig. 60. Also if the descent into them is 

sudden, they may be plainly indi- 
cated by the crest of the high ground before them, together with 
the greater distinctness of the objects shown in the foreground, 
as in the following sketches ; the first a sea view from a precipitous 
shore, the second a land view of a broad valley seen from near the 
crest of an elevated table land. 




112 



LINEAR PERSPECTIVE. 




>*&f 



Fig. 61. 




Fig. 62. 




137. Ascent and descent from the observer, is indicated, in land- 
scape views, by the placing 
of men, animals, shrubs, 
rocks, &c., respectively, fur- 
ther and further above, or 
below the horizon. In 
street views, the relative di- 
rection of the basement and 
sidewalk lines will show 
the same thing. Thus in 
the engraving, V is the van- 
ishing point of horizontal 
lines, and V, of the street 

lines ; hence the view is that of a descending street. See also the 
remote figure, whose eye is below the horizon. 

In the following figure, however, where V, the vanishing point of 
the street lines, is above V, that of the horizontal lines, the street 
evidently ascends. 

138. Level of the eye. In interior and street views, particularly, 
the horizon should not be chosen thoughtlessly, or in improbable 
positions. In Fig. 64, its position indicates a view taken 
either from higher ground than that shown in the figure, or from 
the second story of houses like those on the right, and standing 
on the level of the line aV. 



Fig. 63. 



PICTURES, AND AERIAL PERSPECTIVE. 



113 



\^9. Heflections in water. Two particulars may here be ncted. 
First y reflections of the sun and moon on the Avater should not 

make an acute angle with 
the horizon, as is some- 
times done, but should be 
perpendicular to that line, 
in the picture. The rea- 
son is obvious. The inci- 
dent rays to the water, 
and the reflected ray a 
from it to the eye, by 
which the band of light on 
the water becomes visible, 
are in a vertical plane. 
But the intersection of 
this plane with the per- 
spective plane, is the per- 
spective of this reflection, and this intersection is perpendicular to the 
horizon (79). 

Second ^ images in the water sometimes show surfaces of an ob- 
ject, not seen directly. Thus in the annexed figure of a wall stand- 




FiG. 64. 




Fig. 65. 



ing in water, the top of the coping is seen on the wall itself, but 
the under side in the reflection ; which, being nearest the water, 
can alone send rays to it to be reflected to the eye. 

140. Location of the centre of the picture. This point, in order 
to afford an equally clear view of all parts of the picture, should 
coincide with the geometrical centre of the drawing. It may be 
varied slightly from this position, however, in order to show with 
especial distinctness the more interesting or important parts of the 



114 LINEAR PERSPECTIVE. 

picture. In " bird's eye views," it will be quite above the middle 
of the canvas. 

141. Location of the perspective plane. This has, in all the pre. 
ceding problems, been understood to be between the eye and the 
object. It is not necessarily so, but is so placed in order, first to 
reduce, rather than expand any graphical errors in the projections, 
and, second, to avoid the increased length and confusion of the 
lines of construction, which would result from having the perspec- 
tive larger than the projections. It is evident that the latter con- 
sequence would follow, for the eye being the vertex of the visual 
cone, and the apparent contour of the given object its base, then 
the section of this cone, made by the perspective plane if placed 
beyond the object, would be larger than that base. That is, the 
perspective of the object would be larger than its apparent contour. 
If, then, there is any error in the projections of the object, this er- 
ror will be magnified in the magnified perspective thus produced. 
j^ Illustration. See Fig. 66. Let a- 

i\ , a'h' be a vertical line, at the distance 

I [\ ca in front of the perspective plane : 

I 3.e' and let EE' be the point of sight. Then 

— ^\ '^ I by (58) the visual ray Ea-E'a' pierces 

\j j the perspective plane at A, the per- 

V I spective of aa'., and the ray Ea-E'&' 

\ I pierces it at B, the perspective of ah'. 

'•|k Hence AB is the perspective of a-a'h\ 

\ and is evidently larger than the latter 

j line. 

i 142. Shadows of trees., and other ver- 

FiG, 66. tical objects. Knowing that these are 

parallel in fact, when they fall on the 
same plane, they might, by overlooking Ex. 7, Rem. a be made 
so in the drawing. They should, however, all converge to one 
point, so that in the picture^ shadows of posts, &c., at the right 
of the vanishing point of rays will incline to the left, while siinilar 
shadows on the left of the same point, will incline to the right, 
when the light comes from behind the observer. 

143. Time of a given aspect. By knowing the direction of 
vision, the direction of the shadows of vertical lines upon the 
ground will indicate the part of the day at which a view was taken. 
Thus, if the direction of the shadows indicates that the vanishing 
point of rays, R, is to the right and below the centre of the picture, 
while the observer faces the west, the picture will represent a 



PICTURES, AIS^D AERIAL PERSPECTIVE. 115 

morning scene ; or in summer, and in high latitudes, the same posi« 
tion of R would indicate an early morning view to an observer 
facing the south. These two cases could be distinguished by th 
lengths of the shadows. Also if R were above and to the right of 
the eye, the observer looking south, an afternoon view would be 
indicated. This is a point of some practical importance in reveal 
ng the aspect of dwellings, &c , on proposed sites, at given time 
of day. 

144. Light and shade. The intensity and distribution of shade 
upon a body, depends on so many circumstances, and is subject to 
so many modifications, that its exact representation in real cases 
must be mostly an art of pure imitation. A few points are here 
mentioned, as guides to the observations of the student. 

1. — The light and shade of a body depends upon its form. 
Double curved surfaces (87) have a brilliant pointy or point so 
situated as to reflect the most rays to the eye. Plane and single 
curved surfaces, have a similar brilliant line. On all bodies, the 
line, at all points of which rays are tangent to the body, is the 
apparently darkest line. Its exact construction, in any given case, 
is a problem of practical geometry, of more or less complexity. 

2, — Light and shade, depends upon the nature of a surface, as 
dull or polished. In the former case, the brilliant point is less 
intense and is more expanded into an area, while all parts of the 
body towards the eye are distinctly visible. In the latter case, the 
more perfect the polish, the smaller and more intense the brilliant 
point, and the more nearly invisible all the rest of the body, owing 
to the absence of reflectinos from it, directed towards the eye. 

3. — Lio^ht and shade is afl*ected by distance. If a surface is in 
the light., the more distant it is, the darker it appears, owing to the 
extinguishment of the reflected rays from it by the atmosphere, 
and floating particles therein. If it be in the darJc, the more 
distant it is, the lighter it appears, since we attribute to it the 
increased light entering the eye from the greater dejHh of illu 
mined air between us and it. 

If, then, a surface be seen obliquely, it will appear gradually 
darker as it recedes, if it is in the light ; and gradually lighter as it 
recedes, if it is in the dark. Shadows, in like manner, are darkest 
where nearest to the objects casting them, and lightest in their 
remotest portions. 

Hence, at very great distances, the contrast between light and 
shade diminishes, as seen in the faint shades and shadows of hills 
in the misty distance. 



116 LINEAR PERSPECTIVE. 

4.®— Ligiit and shade is again affected by the nature of the light. 
A diffused light, as on a cloudy day, partially confounds lights 
and shades in a monotonous uniformity. A concentrated light, 
as on a clear day, affords well defined and vivid contrasts of light 
and shade. 

But again ; an intense light, as that of the sun, producer 
reflections so strong as to diminish the contrasts of light an .5 
shade, while the black shadows occasioned by the weaker moonlight 
are in familiar contrast with the white lights afforded by it. 

145. .Edges. In shaded drawings, edges are never to be distin- 
guished by black lines. Being really rounded, through imperfec- 
tion of human instruments, or by attrition or crumbling, they have 
their own brilliant lines, as cylindrical surfaces, and are distinguished 
by lighter tints for a minute width. The brilliant lines of edges in 
the light, are due to primary light falling on them ; those of edges 
in the dark, that is which separate two dark surfaces, to reflected 
light ; and both, to the superior polish acquired in part by the fric- 
tion of passing particles to which edges are exposed. 

Those edges, however, which separate light from shaded plane 
surfaces, are minute cylindrical surfaces so placed as to have a line 
of shade (144, 1°) upon them, and may be indicated by a line of 
slightly darker tint than that of the illuminated surface which they 
bound. 

Hence parallel surfaces, near together, should not be distin- 
guished by material differences of tint on their illuminated portions, 
but by the treatment of their edges, and by the shadows, if any, 
of the foremost on the one behind it. 

146. The Color of objects is modified chiefly by the color oj the 
light by which they are seen ; and by distance, and the condition 
of the atmosphere. To do justice to the former topic, would lead 
further into optical discussions than is here projDOsed. In respect 
to the latter, we observe, that distance causes all colors to be 
confounded more and more in the blue of the atmospheric depths 
through which they are seen. 

Trusting that these problems and notes have now been sufiicient- 
ly extended, to guard the geometrical draftsman from doing offen- 
sive violence to artistic truth, and the artist from doing equally 
offensive violence to geometrical truth, we here terminate both. 



APPENDIX. 



§ 1. — Description of Plates. 

Photography has so far replaced perspective as a means of 
representing regular objects of all kinds, already existing, that, 
however useful perspectives may be for mental discipline as a 
branch of applied geometry, its chief practical application is to 
architectural drawing, whether of exteriors or interiors, of build- 
ings yet to be constructed. 

We therefore conclude with two illustrative plates of exam- 
ples of such work as can be done by the methods now explained 
in this volume, employing those, either of Part I. or Part II., 
as may seem most simple and convenient to the draughtsman. 

To insure good pictorial effect, the eye must be supposed to 
be at a suitable distance from the drawing ; at from ten to thirty 
inches, according to the scale of the drawing. To make this 
possible, otherwise than by the modifications explained in my 
''^ Higher Perspective," and in the portion on Perspective, of 
my ^^ Elements of Descriptive Geometry," the paper should be 
pinned to a drawing board or table large enough to allow 
the vanishing points to be outside the limits of the sheet. 
Then with the long straight edges common in all draughting 
offices, and with needles set up at the vanishing points, there 
will be no difficulty in making drawings of large size and good 
pictorial effect. For pupils' purposes, ordinary instruments and 
desks will be entirely sufficient ; the paper being fastened in 
tlie middle of the drawing board or desk. 

The first figure on Plate I. shows a building in dwelling- 
house style. To find its plan from its given perspective."^ — 
By noting where any three (taking the third as a check) hor- 
izontal lines of the visible end of the main building meet, 
we should have V, their vanishing point, for one point of the 

* This cau be omitted at the discretiou of teachers. 



118 APPENDIX. 

horizon (106). Proceeding in like manner with any three 
horizontal lines of the front of the building, we should find Vi, 
their vanishing point, as a second point of the horizon, which 
line, VVi, can then be drawn. 

Next, on VVi as a diameter, describe a semicircle, and the 
position of the eye itself will be somewhere on this semicircle, 
as S, Fig. 49, is evidently on a semicircle, having LY for its 
diameter. Now notice that the left-hand rear vertical line of 
the building, which we will call A, is longer, though but little 
longer, than the extreme right-hand vertical line, B, of the 
building. This shows that the former line is a little, but only 
a little, nearer to the perspective plane than the latter one is. 
Now, draw lines from the foot of A to Vi, and from the foot of 
B to V. These, with the visible bottom lines of the building, 
will give the complete perspective of the base of the building. 
Draw that diagonal of this perspective which meets the horizon 
between V and Vi, at a point which we will call M, and which 
will be the vanishing point of that base diagonal. 

We can now assume a position, E, for the eye on the semi- 
circle on VVi above described, and draw EV, EVi, and EM. 
These will (Fig. 49) show the true directions of the sides, and the 
one diagonal mentioned of the plan of the building, and will fix 
its proportions. By then drawing a plan having its sides in these 
directions, and located as may be learned by inspecting Fig. 49 
(where see VL', L'L, and LM ; or Vw", /7i"m, and mM), we 
shall see by one or two trials whether we have assumed E so as 
to make line A, as it should be, nearer to the ground line than 
line B. 

Otherwise, in both plates, assume at pleasure plans which 
will give perspectives resembling those shown, in their character- 
istic geometrical features ; that is, mostly, with few, large, and 
simple lines. 

In Fig. 48, by completing the perspective of the base of the 
tower, and by drawing the diagonals of that perspective base, 
their intersection will be the perspective of the centre of that 
base. The apex, a, of the spire will then be in a vertical line 
from that centre. Such an auxiliary, with either a diagonal or a 
perpendicular, can be used in finding the apex of the church 
tower in Plate I. 

Whenever, as in Plate XL, the building stands on level ground. 



APPENDIX. 



119 



the horizon may be assumed to be at the level of the eye of a 
person standing on the ground. But if, for example, the bands 
on the tower, Plate I., Fig. 2, should meet, if produced, above 
or below the level of the eye of a person in the doorway, it would 
show that the spectator stood above or below the other's level. 

Plate II. affords examples of circular lines in various positions, 
horizontal and vertical (Ex. 10, Fig. 55) ; and note that the 
perspective of a horizontal circle can be found by diagonals and 
perpendiculars through a few principal points, as in Examples 4 
and 8 ; only being careful to consider the height of the circle, 
as was done with the upper base of the tower in Example 4. 
See also Example 12, in constructing the right-hand gables. 

After a little practice, minor details can be sketched in, after 
constructing the governing or main lines of the object. 

The perspectives of gothic interiors afford a good illustra- 
tion of this remark. Thus, 
let Fig. 67 represent the 
plan of a single compart- 
ment, or bay, of a long gothic 
interior. The lines ab and 
cd are then the plans of the 
intersections of the longi- 
tudinal and transverse arches 
whose topmost lines are AB 
and CD,* and the squares at 
a, h, c, and d are the plans 
of the piers on which the 
arches rest. Then, by find- 
ing the perspectives of the few governing points indicated in the 
plan (taking their heights from an elevation), the moulded capi- 
tals, ribs, and ^^ bosses " at the intersections can be sketched in. 




§ 2. — Generalization of the Horizontal Plane, and Horizon. 

The horizontal plane is simply one of any number of planes, 
all of them alike in being perpendicular to the perspective 
plane. Hence, each of these planes will have a line having the 
same relation to it that the horizon has to a horizontal plane. 

* Note, however, that ah and cd, in plan view, will not be straight, unless ac 
and ad are equal, as may be seen by careful inspection of the interiors themselves ; 
or unless they are assumed as straight, which will then determine the form of the 
curve of one of the cross sections, ac or ad, the other being first assumed. 



130 



APPENDIX. 



Thus, in Fig. 68, let the paper be the perspective plane, sup-, 
posed to be in a vertical position. Then, if line A is the trace, 
upon the perspective plane, of any horizontal plane, and E the 
projection of the point of sight upon the perspective plane, the 
line ED, parallel to A, is the horizon. Then, if ES is the per- 
pendicular distance of the eye, in space, in front of E, and if 
ED = ES, then D is a vanishing point of diagonals (112-114). 

Likewise, if the line B be the trace of a vertical plane, and 
the line that of an oblique plane, both perpendicular to the 
perspective plane, then the lines EDi, parallel to B, and ED2, 
parallel to C, are the same thing to those planes, respectively, 

that ED is to the 
horizontal plane A. 
Hence EDi may be 
spoken of as the hori- 
zon of the plane B, 
and ED2, as that of 
the plane 0. More- 
over, as all lines 
7^ through the one fixed 
point of sight at the 
distance ES, in front 
of the perspective 
plane, and making 
angles of 45° with 
that plane, will meet 
it at equal distances 
from E, describe a 
circle with radius ED 
(=ES) and Di, on 
EDi ; and Da, on EDj, will be the vanishing points of such lines, 
parallel to plane B, and plane 0, respectively ; just as D is a 
vanishing point of horizontal diagonals. These lines, indeed, 
are diagonals relative to their respective planes. 

Accordingly, if aa' be a point in plane A, at the distance aa' 
heJiind the perspective plane, and if this point be shown in the 
paper at «, by revolving that portion of plane A which is behind 
the perspective plane, outwardly from E, then aa' is the per- 
pendicular and ad the diagonal from the point aa'. Then «'E 
and dT), the perspectives of this perpendicular and this diagonal, 
will intersect at p, which will be the perspective of the point aa'. 




■Fis^ 68 



APPENDIX. 121 

Proceeding similarly, in every respect, with the point W , in the 
plane B ; and the point cc' , in the plane C, as fully shown in 
the figure, we find p\, the perspective of the point hV ; and jpi 
the perspective of the point cc' , 

Many problems may be very simply and neatly solved by an 
application of the method of this section. For example, if the 
building shown in the first figure of Plate I. were placed with 
its front parallel to the perspective plane, all its horizontal 
planes would be like plane A, Fig. 68 ; the ends of the building 
would be planes like plane B ; and the end slopes of its hipped 
roof would be such planes as — that is, perpendicular to the 
perspective plane, but oblique to the horizontal plane. 

§ 3. — Perspectives from ^^Perspective Plans .'^ 

Let APBE (Fig 47) represent the plan, forming one of a 
series of working drawings of an architectural structure. The 
perspective of every point of this plan. A, F, P, etc., consid- 
ered now simply as a figure in the horizontal plane, can then 
readily be found, just as the points p and e were. Then, tem- 
porarily fastening such a complete perspective plan, as it is 
called, to the lower edge of the sheet which is to contain the 
perspective view of the structure, this view can be rapidly made 
by drawing perpendiculars to the ground line from the points 
of the perspective plan, limited at the foot by transferring the 
distances of the points of the perspective plan from its horizon. 
Then all points in these perpendiculars which, in space, are in 
the same horizontal plane can be limited at once, as the verticals 
at/and^are in Fig. 49, or those at B' and e are in Fig. 47. 

This method is very useful in architectural practice. 

The perspective view of a building being thus made, its 
principal shadows can then be found by the method of Pakt II., 
. Chapter II., and illustrated in subsequent problems. 



THE END. 



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